Page 1
On approximate Birkhoff orthogonality in normed spaces
Jacek Chmielinski
Instytut MatematykiUniwersytet Pedagogiczny w Krakowie
Banach Spaces and their ApplicationsLviv (Ukraine), June 26-29, 2019
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 1 / 28
Page 2
Introduction — inner product space
(X , 〈·|·〉) — inner product space; x⊥y ⇔ 〈x |y〉 = 0.
Approximate orthogonality (ε-orthogonality with ε ∈ [0, 1)):
x⊥ε y ⇔ | 〈x |y〉 | ≤ ε ‖x‖ ‖y‖, x , y ∈ X .
Observation
x⊥ε y ⇔ ∃ z ∈ X : x⊥z , ‖z − y‖ ≤ ε‖y‖.
Indeed, if x⊥ε y take z = − 〈x |y〉‖x‖2 x + y (z = y in case x = 0).
Conversely, assuming x⊥z and ‖z − y‖ ≤ ε‖y‖,
| 〈x |y〉 | = | 〈x |y − z〉 | ≤ ‖x‖ ‖y − z‖ ≤ ε‖x‖ ‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 2 / 28
Page 3
Introduction — inner product space
(X , 〈·|·〉) — inner product space; x⊥y ⇔ 〈x |y〉 = 0.
Approximate orthogonality (ε-orthogonality with ε ∈ [0, 1)):
x⊥ε y ⇔ | 〈x |y〉 | ≤ ε ‖x‖ ‖y‖, x , y ∈ X .
Observation
x⊥ε y ⇔ ∃ z ∈ X : x⊥z , ‖z − y‖ ≤ ε‖y‖.
Indeed, if x⊥ε y take z = − 〈x |y〉‖x‖2 x + y (z = y in case x = 0).
Conversely, assuming x⊥z and ‖z − y‖ ≤ ε‖y‖,
| 〈x |y〉 | = | 〈x |y − z〉 | ≤ ‖x‖ ‖y − z‖ ≤ ε‖x‖ ‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 2 / 28
Page 4
Introduction — inner product space
(X , 〈·|·〉) — inner product space; x⊥y ⇔ 〈x |y〉 = 0.
Approximate orthogonality (ε-orthogonality with ε ∈ [0, 1)):
x⊥ε y ⇔ | 〈x |y〉 | ≤ ε ‖x‖ ‖y‖, x , y ∈ X .
Observation
x⊥ε y ⇔ ∃ z ∈ X : x⊥z , ‖z − y‖ ≤ ε‖y‖.
Indeed, if x⊥ε y take z = − 〈x |y〉‖x‖2 x + y (z = y in case x = 0).
Conversely, assuming x⊥z and ‖z − y‖ ≤ ε‖y‖,
| 〈x |y〉 | = | 〈x |y − z〉 | ≤ ‖x‖ ‖y − z‖ ≤ ε‖x‖ ‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 2 / 28
Page 5
Introduction — inner product space
(X , 〈·|·〉) — inner product space; x⊥y ⇔ 〈x |y〉 = 0.
Approximate orthogonality (ε-orthogonality with ε ∈ [0, 1)):
x⊥ε y ⇔ | 〈x |y〉 | ≤ ε ‖x‖ ‖y‖, x , y ∈ X .
Observation
x⊥ε y ⇔ ∃ z ∈ X : x⊥z , ‖z − y‖ ≤ ε‖y‖.
Indeed, if x⊥ε y take z = − 〈x |y〉‖x‖2 x + y (z = y in case x = 0).
Conversely, assuming x⊥z and ‖z − y‖ ≤ ε‖y‖,
| 〈x |y〉 | = | 〈x |y − z〉 | ≤ ‖x‖ ‖y − z‖ ≤ ε‖x‖ ‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 2 / 28
Page 6
Introduction — inner product space
(X , 〈·|·〉) — inner product space; x⊥y ⇔ 〈x |y〉 = 0.
Approximate orthogonality (ε-orthogonality with ε ∈ [0, 1)):
x⊥ε y ⇔ | 〈x |y〉 | ≤ ε ‖x‖ ‖y‖, x , y ∈ X .
Observation
x⊥ε y ⇔ ∃ z ∈ X : x⊥z , ‖z − y‖ ≤ ε‖y‖.
Indeed, if x⊥ε y take z = − 〈x |y〉‖x‖2 x + y (z = y in case x = 0).
Conversely, assuming x⊥z and ‖z − y‖ ≤ ε‖y‖,
| 〈x |y〉 | = | 〈x |y − z〉 | ≤ ‖x‖ ‖y − z‖ ≤ ε‖x‖ ‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 2 / 28
Page 7
Introduction — inner product space
(X , 〈·|·〉) — inner product space; x⊥y ⇔ 〈x |y〉 = 0.
Approximate orthogonality (ε-orthogonality with ε ∈ [0, 1)):
x⊥ε y ⇔ | 〈x |y〉 | ≤ ε ‖x‖ ‖y‖, x , y ∈ X .
Observation
x⊥ε y ⇔ ∃ z ∈ X : x⊥z , ‖z − y‖ ≤ ε‖y‖.
Indeed, if x⊥ε y take z = − 〈x |y〉‖x‖2 x + y (z = y in case x = 0).
Conversely, assuming x⊥z and ‖z − y‖ ≤ ε‖y‖,
| 〈x |y〉 | = | 〈x |y − z〉 | ≤ ‖x‖ ‖y − z‖ ≤ ε‖x‖ ‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 2 / 28
Page 8
Birkhoff orthogonality
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 3 / 28
Page 9
Birkhoff orthogonalityG. Birkhoff, Orthogonality in linear metric spaces. Duke Math. J., 1 (1935), 169–172.
(X , ‖ · ‖) a real normed space.
x⊥By :⇐⇒ ∀λ ∈ R : ‖x + λy‖ ≥ ‖x‖.
xy
x+λy
Figure: R2 with the maximum norm; x⊥By
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 4 / 28
Page 10
Birkhoff orthogonalityG. Birkhoff, Orthogonality in linear metric spaces. Duke Math. J., 1 (1935), 169–172.
(X , ‖ · ‖) a real normed space.
x⊥By :⇐⇒ ∀λ ∈ R : ‖x + λy‖ ≥ ‖x‖.
xy
x+λy
Figure: R2 with the maximum norm; x⊥By
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 4 / 28
Page 11
Birkhoff orthogonalityG. Birkhoff, Orthogonality in linear metric spaces. Duke Math. J., 1 (1935), 169–172.
(X , ‖ · ‖) a real normed space.
x⊥By :⇐⇒ ∀λ ∈ R : ‖x + λy‖ ≥ ‖x‖.
xy
x+λy
Figure: R2 with the maximum norm; x⊥By
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 4 / 28
Page 12
Birkhoff orthogonalityG. Birkhoff, Orthogonality in linear metric spaces. Duke Math. J., 1 (1935), 169–172.
(X , ‖ · ‖) a real normed space.
x⊥By :⇐⇒ ∀λ ∈ R : ‖x + λy‖ ≥ ‖x‖.
xy
x+λy
Figure: R2 with the maximum norm; x⊥By
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 4 / 28
Page 13
Approximate Birkhoff orthogonality
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 5 / 28
Page 14
Approximate Birkhoff orthogonality
For ε ∈ [0, 1) we consider an ε-Birkhoff orthogonality ⊥εB.
J. Chmielinski, On an ε-Birkhoff orthogonality, J. Inequal. Pure andAppl. Math. 6 (2005), Art. 79.
x⊥εBy :⇐⇒ ∀λ ∈ K : ‖x + λy‖2 ≥ ‖x‖2 − 2ε‖x‖ ‖λy‖.
J. Chmielinski, T. Stypu la, P. Wojcik, Approximate orthogonality innormed spaces and its applications, Linear Algebra and itsApplications 531 (2017), 305–317.
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 6 / 28
Page 15
Approximate Birkhoff orthogonality
For ε ∈ [0, 1) we consider an ε-Birkhoff orthogonality ⊥εB.
J. Chmielinski, On an ε-Birkhoff orthogonality, J. Inequal. Pure andAppl. Math. 6 (2005), Art. 79.
x⊥εBy :⇐⇒ ∀λ ∈ K : ‖x + λy‖2 ≥ ‖x‖2 − 2ε‖x‖ ‖λy‖.
J. Chmielinski, T. Stypu la, P. Wojcik, Approximate orthogonality innormed spaces and its applications, Linear Algebra and itsApplications 531 (2017), 305–317.
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 6 / 28
Page 16
Approximate Birkhoff orthogonality
For ε ∈ [0, 1) we consider an ε-Birkhoff orthogonality ⊥εB.
J. Chmielinski, On an ε-Birkhoff orthogonality, J. Inequal. Pure andAppl. Math. 6 (2005), Art. 79.
x⊥εBy :⇐⇒ ∀λ ∈ K : ‖x + λy‖2 ≥ ‖x‖2 − 2ε‖x‖ ‖λy‖.
J. Chmielinski, T. Stypu la, P. Wojcik, Approximate orthogonality innormed spaces and its applications, Linear Algebra and itsApplications 531 (2017), 305–317.
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 6 / 28
Page 17
Approximate Birkhoff orthogonality
For ε ∈ [0, 1) we consider an ε-Birkhoff orthogonality ⊥εB.
J. Chmielinski, On an ε-Birkhoff orthogonality, J. Inequal. Pure andAppl. Math. 6 (2005), Art. 79.
x⊥εBy :⇐⇒ ∀λ ∈ K : ‖x + λy‖2 ≥ ‖x‖2 − 2ε‖x‖ ‖λy‖.
J. Chmielinski, T. Stypu la, P. Wojcik, Approximate orthogonality innormed spaces and its applications, Linear Algebra and itsApplications 531 (2017), 305–317.
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 6 / 28
Page 18
Approximate Birkhoff orthogonality
For ε ∈ [0, 1) we consider an ε-Birkhoff orthogonality ⊥εB.
J. Chmielinski, On an ε-Birkhoff orthogonality, J. Inequal. Pure andAppl. Math. 6 (2005), Art. 79.
x⊥εBy :⇐⇒ ∀λ ∈ K : ‖x + λy‖2 ≥ ‖x‖2 − 2ε‖x‖ ‖λy‖.
J. Chmielinski, T. Stypu la, P. Wojcik, Approximate orthogonality innormed spaces and its applications, Linear Algebra and itsApplications 531 (2017), 305–317.
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 6 / 28
Page 19
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
1−1
1
−1
x
y
z
Figure: R2 with l∞-l1-norm
x 6⊥By , x⊥z , ‖z − y‖ ≤ ε⇒ x⊥εBy .
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 7 / 28
Page 20
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
1−1
1
−1
x
y
z
Figure: R2 with l∞-l1-norm
x 6⊥By , x⊥z , ‖z − y‖ ≤ ε⇒ x⊥εBy .
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 7 / 28
Page 21
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
1−1
1
−1
x
y
z
Figure: R2 with l∞-l1-norm
x 6⊥By
, x⊥z , ‖z − y‖ ≤ ε⇒ x⊥εBy .
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 7 / 28
Page 22
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
1−1
1
−1
x
y
z
Figure: R2 with l∞-l1-norm
x 6⊥By , x⊥z
, ‖z − y‖ ≤ ε⇒ x⊥εBy .
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 7 / 28
Page 23
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
1−1
1
−1
x
y
z
Figure: R2 with l∞-l1-norm
x 6⊥By , x⊥z , ‖z − y‖ ≤ ε
⇒ x⊥εBy .
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 7 / 28
Page 24
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
1−1
1
−1
x
y
z
Figure: R2 with l∞-l1-norm
x 6⊥By , x⊥z , ‖z − y‖ ≤ ε⇒ x⊥εBy .
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 7 / 28
Page 25
For 0 6= x ∈ X we consider the class of its supporting functionals:
J(x) = {ϕ ∈ X ∗ : ‖ϕ‖ = 1, ϕ(x) = ‖x‖ }.
Theorem
Let X be a real normed space, x , y ∈ X and ε ∈ [0, 1). Then
x⊥εBy ⇔ ∃ϕ ∈ J(x) : |ϕ(y)| ≤ ε‖y‖.
In particular (James),
x⊥By ⇔ ∃ϕ ∈ J(x) : ϕ(y) = 0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 8 / 28
Page 26
For 0 6= x ∈ X we consider the class of its supporting functionals:
J(x) = {ϕ ∈ X ∗ : ‖ϕ‖ = 1, ϕ(x) = ‖x‖ }.
Theorem
Let X be a real normed space, x , y ∈ X and ε ∈ [0, 1). Then
x⊥εBy ⇔ ∃ϕ ∈ J(x) : |ϕ(y)| ≤ ε‖y‖.
In particular (James),
x⊥By ⇔ ∃ϕ ∈ J(x) : ϕ(y) = 0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 8 / 28
Page 27
For 0 6= x ∈ X we consider the class of its supporting functionals:
J(x) = {ϕ ∈ X ∗ : ‖ϕ‖ = 1, ϕ(x) = ‖x‖ }.
Theorem
Let X be a real normed space, x , y ∈ X and ε ∈ [0, 1). Then
x⊥εBy ⇔ ∃ϕ ∈ J(x) : |ϕ(y)| ≤ ε‖y‖.
In particular (James),
x⊥By ⇔ ∃ϕ ∈ J(x) : ϕ(y) = 0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 8 / 28
Page 28
Applications
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 9 / 28
Page 29
Orthogonality of operators on a Hilbert space
H – Hilbert space; L(H) – the space of linear bounded operators on H.For T ∈ L(H):
MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
R. Bhatia, P. Semrl, Orthogonality of matrices and some distanceproblems, Linear Algebra Appl. 287 (1999), 77-85.
Theorem (Bhatia-Semrl)
Let H be a Hilbert space and let T ,S ∈ L(H). Then, the followingconditions are equivalent:
(1) T⊥BS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, 〈Txn|Sxn〉 → 0 (n→∞).
Moreover, if dimH <∞ and T ,S ∈ L(H), then each of the aboveconditions is equivalent to:
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥Sx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 10 / 28
Page 30
Orthogonality of operators on a Hilbert space
H – Hilbert space; L(H) – the space of linear bounded operators on H.For T ∈ L(H):
MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
R. Bhatia, P. Semrl, Orthogonality of matrices and some distanceproblems, Linear Algebra Appl. 287 (1999), 77-85.
Theorem (Bhatia-Semrl)
Let H be a Hilbert space and let T ,S ∈ L(H). Then, the followingconditions are equivalent:
(1) T⊥BS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, 〈Txn|Sxn〉 → 0 (n→∞).
Moreover, if dimH <∞ and T ,S ∈ L(H), then each of the aboveconditions is equivalent to:
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥Sx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 10 / 28
Page 31
Orthogonality of operators on a Hilbert space
H – Hilbert space; L(H) – the space of linear bounded operators on H.For T ∈ L(H):
MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
R. Bhatia, P. Semrl, Orthogonality of matrices and some distanceproblems, Linear Algebra Appl. 287 (1999), 77-85.
Theorem (Bhatia-Semrl)
Let H be a Hilbert space and let T ,S ∈ L(H). Then, the followingconditions are equivalent:
(1) T⊥BS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, 〈Txn|Sxn〉 → 0 (n→∞).
Moreover, if dimH <∞ and T ,S ∈ L(H), then each of the aboveconditions is equivalent to:
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥Sx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 10 / 28
Page 32
Orthogonality of operators on a Hilbert space
H – Hilbert space; L(H) – the space of linear bounded operators on H.For T ∈ L(H):
MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
R. Bhatia, P. Semrl, Orthogonality of matrices and some distanceproblems, Linear Algebra Appl. 287 (1999), 77-85.
Theorem (Bhatia-Semrl)
Let H be a Hilbert space and let T , S ∈ L(H). Then, the followingconditions are equivalent:
(1) T⊥BS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, 〈Txn|Sxn〉 → 0 (n→∞).
Moreover, if dimH <∞ and T ,S ∈ L(H), then each of the aboveconditions is equivalent to:
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥Sx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 10 / 28
Page 33
Approximate orthogonality in L(H)
H – Hilbert space; T ∈ L(H); MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
Theorem
For T ,S ∈ L(H) the following conditions are equivalent:
(1) T⊥εBS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, limn→∞ | 〈Txn|Sxn〉 | ≤ ε‖T‖ ‖S‖.
Moreover, if dimH <∞, then the above conditions are equivalent to
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, | 〈Tx0|Sx0〉 | ≤ ε‖T‖ ‖S‖.
If dimH <∞ or if T is compact and if additionally MT ⊂ MS , the abovethree conditions are equivalent also to
(4) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥εSx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 11 / 28
Page 34
Approximate orthogonality in L(H)
H – Hilbert space; T ∈ L(H); MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
Theorem
For T ,S ∈ L(H) the following conditions are equivalent:
(1) T⊥εBS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, limn→∞ | 〈Txn|Sxn〉 | ≤ ε‖T‖ ‖S‖.
Moreover, if dimH <∞, then the above conditions are equivalent to
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, | 〈Tx0|Sx0〉 | ≤ ε‖T‖ ‖S‖.
If dimH <∞ or if T is compact and if additionally MT ⊂ MS , the abovethree conditions are equivalent also to
(4) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥εSx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 11 / 28
Page 35
Approximate orthogonality in L(H)
H – Hilbert space; T ∈ L(H); MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
Theorem
For T , S ∈ L(H) the following conditions are equivalent:
(1) T⊥εBS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, limn→∞ | 〈Txn|Sxn〉 | ≤ ε‖T‖ ‖S‖.
Moreover, if dimH <∞, then the above conditions are equivalent to
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, | 〈Tx0|Sx0〉 | ≤ ε‖T‖ ‖S‖.
If dimH <∞ or if T is compact and if additionally MT ⊂ MS , the abovethree conditions are equivalent also to
(4) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥εSx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 11 / 28
Page 36
Approximate orthogonality in L(H)
H – Hilbert space; T ∈ L(H); MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
Theorem
For T , S ∈ L(H) the following conditions are equivalent:
(1) T⊥εBS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, limn→∞ | 〈Txn|Sxn〉 | ≤ ε‖T‖ ‖S‖.
Moreover, if dimH <∞, then the above conditions are equivalent to
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, | 〈Tx0|Sx0〉 | ≤ ε‖T‖ ‖S‖.
If dimH <∞ or if T is compact and if additionally MT ⊂ MS , the abovethree conditions are equivalent also to
(4) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥εSx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 11 / 28
Page 37
Approximate orthogonality in L(H)
H – Hilbert space; T ∈ L(H); MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
Theorem
For T , S ∈ L(H) the following conditions are equivalent:
(1) T⊥εBS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, limn→∞ | 〈Txn|Sxn〉 | ≤ ε‖T‖ ‖S‖.
Moreover, if dimH <∞, then the above conditions are equivalent to
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, | 〈Tx0|Sx0〉 | ≤ ε‖T‖ ‖S‖.
If dimH <∞ or if T is compact and if additionally MT ⊂ MS , the abovethree conditions are equivalent also to
(4) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥εSx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 11 / 28
Page 38
Approximate orthogonality in L(H)
H – Hilbert space; T ∈ L(H); MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
Theorem
For T , S ∈ L(H) the following conditions are equivalent:
(1) T⊥εBS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, limn→∞ | 〈Txn|Sxn〉 | ≤ ε‖T‖ ‖S‖.
Moreover, if dimH <∞, then the above conditions are equivalent to
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, | 〈Tx0|Sx0〉 | ≤ ε‖T‖ ‖S‖.
If dimH <∞ or if T is compact and if additionally MT ⊂ MS , the abovethree conditions are equivalent also to
(4) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥εSx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 11 / 28
Page 39
Approximate orthogonality in L(H)
H – Hilbert space; T ∈ L(H); MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
Theorem
For T , S ∈ L(H) the following conditions are equivalent:
(1) T⊥εBS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, limn→∞ | 〈Txn|Sxn〉 | ≤ ε‖T‖ ‖S‖.
Moreover, if dimH <∞, then the above conditions are equivalent to
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, | 〈Tx0|Sx0〉 | ≤ ε‖T‖ ‖S‖.
If dimH <∞ or if T is compact and if additionally MT ⊂ MS , the abovethree conditions are equivalent also to
(4) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥εSx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 11 / 28
Page 40
Approximate orthogonality in C0(K )
L(H) Let K be a locally compact topological space.
C0(K ) := {f : K → R cont. : ∀ ε > 0, {t ∈ K : |f (t)| ≥ ε} compact}
– with the supremum norm. For f ∈ C0(K ), Mf := {t ∈ K : |f (t)| = ‖f ‖}(nonempty and compact).
Theorem
Let f , g ∈ C0(K ), f 6= 0 6= g . Assume that Mf is connected. Then, thefollowing conditions are equivalent:
(a) f⊥εBg ,
(b) ∃ t1 ∈ Mf : |g(t1)| ≤ ε‖g‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 12 / 28
Page 41
Approximate orthogonality in C0(K )
L(H) Let K be a locally compact topological space.
C0(K ) := {f : K → R cont. : ∀ ε > 0, {t ∈ K : |f (t)| ≥ ε} compact}
– with the supremum norm. For f ∈ C0(K ), Mf := {t ∈ K : |f (t)| = ‖f ‖}(nonempty and compact).
Theorem
Let f , g ∈ C0(K ), f 6= 0 6= g . Assume that Mf is connected. Then, thefollowing conditions are equivalent:
(a) f⊥εBg ,
(b) ∃ t1 ∈ Mf : |g(t1)| ≤ ε‖g‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 12 / 28
Page 42
Approximate orthogonality in C0(K )
L(H) Let K be a locally compact topological space.
C0(K ) := {f : K → R cont. : ∀ ε > 0, {t ∈ K : |f (t)| ≥ ε} compact}
– with the supremum norm. For f ∈ C0(K ), Mf := {t ∈ K : |f (t)| = ‖f ‖}(nonempty and compact).
Theorem
Let f , g ∈ C0(K ), f 6= 0 6= g . Assume that Mf is connected. Then, thefollowing conditions are equivalent:
(a) f⊥εBg ,
(b) ∃ t1 ∈ Mf : |g(t1)| ≤ ε‖g‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 12 / 28
Page 43
Approximate orthogonality in C0(K )
L(H) Let K be a locally compact topological space.
C0(K ) := {f : K → R cont. : ∀ ε > 0, {t ∈ K : |f (t)| ≥ ε} compact}
– with the supremum norm. For f ∈ C0(K ), Mf := {t ∈ K : |f (t)| = ‖f ‖}(nonempty and compact).
Theorem
Let f , g ∈ C0(K ), f 6= 0 6= g . Assume that Mf is connected. Then, thefollowing conditions are equivalent:
(a) f⊥εBg ,
(b) ∃ t1 ∈ Mf : |g(t1)| ≤ ε‖g‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 12 / 28
Page 44
Approximate orthogonality in C0(K )
L(H) Let K be a locally compact topological space.
C0(K ) := {f : K → R cont. : ∀ ε > 0, {t ∈ K : |f (t)| ≥ ε} compact}
– with the supremum norm. For f ∈ C0(K ), Mf := {t ∈ K : |f (t)| = ‖f ‖}(nonempty and compact).
Theorem
Let f , g ∈ C0(K ), f 6= 0 6= g . Assume that Mf is connected. Then, thefollowing conditions are equivalent:
(a) f⊥εBg ,
(b) ∃ t1 ∈ Mf : |g(t1)| ≤ ε‖g‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 12 / 28
Page 45
Approximate orthogonality in C0(K )
L(H) Let K be a locally compact topological space.
C0(K ) := {f : K → R cont. : ∀ ε > 0, {t ∈ K : |f (t)| ≥ ε} compact}
– with the supremum norm. For f ∈ C0(K ), Mf := {t ∈ K : |f (t)| = ‖f ‖}(nonempty and compact).
Theorem
Let f , g ∈ C0(K ), f 6= 0 6= g . Assume that Mf is connected. Then, thefollowing conditions are equivalent:
(a) f⊥εBg ,
(b) ∃ t1 ∈ Mf : |g(t1)| ≤ ε‖g‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 12 / 28
Page 46
Approximate symmetry of B-orthogonality
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 13 / 28
Page 47
Symmetry of ⊥B
Birkhoff orthogonality ⊥B is generally not symmetric.
x
y
x+λy
y+λx
Figure: R2 with the maximum norm; x⊥By , y 6⊥Bx
If dimX ≥ 3 and ⊥B – symmetric, then X is an inner product space.If dimX = 2, then the symmetry of ⊥B is possible even if the norm doesnot come from an inner product (Radon plane).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 14 / 28
Page 48
Symmetry of ⊥B
Birkhoff orthogonality ⊥B is generally not symmetric.
x
y
x+λy
y+λx
Figure: R2 with the maximum norm; x⊥By , y 6⊥Bx
If dimX ≥ 3 and ⊥B – symmetric, then X is an inner product space.If dimX = 2, then the symmetry of ⊥B is possible even if the norm doesnot come from an inner product (Radon plane).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 14 / 28
Page 49
Symmetry of ⊥B
Birkhoff orthogonality ⊥B is generally not symmetric.
x
y
x+λy
y+λx
Figure: R2 with the maximum norm; x⊥By , y 6⊥Bx
If dimX ≥ 3 and ⊥B – symmetric, then X is an inner product space.If dimX = 2, then the symmetry of ⊥B is possible even if the norm doesnot come from an inner product (Radon plane).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 14 / 28
Page 50
Symmetry of ⊥B
Birkhoff orthogonality ⊥B is generally not symmetric.
x
y
x+λy
y+λx
Figure: R2 with the maximum norm; x⊥By , y 6⊥Bx
If dimX ≥ 3 and ⊥B – symmetric, then X is an inner product space.
If dimX = 2, then the symmetry of ⊥B is possible even if the norm doesnot come from an inner product (Radon plane).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 14 / 28
Page 51
Symmetry of ⊥B
Birkhoff orthogonality ⊥B is generally not symmetric.
x
y
x+λy
y+λx
Figure: R2 with the maximum norm; x⊥By , y 6⊥Bx
If dimX ≥ 3 and ⊥B – symmetric, then X is an inner product space.If dimX = 2, then the symmetry of ⊥B is possible even if the norm doesnot come from an inner product (Radon plane).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 14 / 28
Page 52
Approximate symmetry of ⊥B
J. Chmielinski, P. Wojcik, Approximate symmetry of the Birkhofforthogonality, J. Math. Anal. Appl. 461 (2018), 625–640.
Definition
The Birkhoff orthogonality relation in a normed space X is calledε-symmetric (for some ε ∈ [0, 1)), if for any x , y ∈ X :
x⊥By =⇒ y⊥εBx .
⊥B is ε-symmetric for some ε ∈ [0, 1) if and only if:
x⊥By =⇒ ∃ z ∈ Lin {x , y} : y⊥Bz , ‖z − x‖ ≤ ε‖x‖.
The approximate symmetry of ⊥B does not imply that the norm comesfrom an inner product (even if dimX ≥ 3).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 15 / 28
Page 53
Approximate symmetry of ⊥B
J. Chmielinski, P. Wojcik, Approximate symmetry of the Birkhofforthogonality, J. Math. Anal. Appl. 461 (2018), 625–640.
Definition
The Birkhoff orthogonality relation in a normed space X is calledε-symmetric (for some ε ∈ [0, 1)), if for any x , y ∈ X :
x⊥By =⇒ y⊥εBx .
⊥B is ε-symmetric for some ε ∈ [0, 1) if and only if:
x⊥By =⇒ ∃ z ∈ Lin {x , y} : y⊥Bz , ‖z − x‖ ≤ ε‖x‖.
The approximate symmetry of ⊥B does not imply that the norm comesfrom an inner product (even if dimX ≥ 3).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 15 / 28
Page 54
Approximate symmetry of ⊥B
J. Chmielinski, P. Wojcik, Approximate symmetry of the Birkhofforthogonality, J. Math. Anal. Appl. 461 (2018), 625–640.
Definition
The Birkhoff orthogonality relation in a normed space X is calledε-symmetric (for some ε ∈ [0, 1)), if for any x , y ∈ X :
x⊥By =⇒ y⊥εBx .
⊥B is ε-symmetric for some ε ∈ [0, 1) if and only if:
x⊥By =⇒ ∃ z ∈ Lin {x , y} : y⊥Bz , ‖z − x‖ ≤ ε‖x‖.
The approximate symmetry of ⊥B does not imply that the norm comesfrom an inner product (even if dimX ≥ 3).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 15 / 28
Page 55
Approximate symmetry of ⊥B
J. Chmielinski, P. Wojcik, Approximate symmetry of the Birkhofforthogonality, J. Math. Anal. Appl. 461 (2018), 625–640.
Definition
The Birkhoff orthogonality relation in a normed space X is calledε-symmetric (for some ε ∈ [0, 1)), if for any x , y ∈ X :
x⊥By =⇒ y⊥εBx .
⊥B is ε-symmetric for some ε ∈ [0, 1) if and only if:
x⊥By =⇒ ∃ z ∈ Lin {x , y} : y⊥Bz , ‖z − x‖ ≤ ε‖x‖.
The approximate symmetry of ⊥B does not imply that the norm comesfrom an inner product (even if dimX ≥ 3).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 15 / 28
Page 56
Approximate symmetry of ⊥B
J. Chmielinski, P. Wojcik, Approximate symmetry of the Birkhofforthogonality, J. Math. Anal. Appl. 461 (2018), 625–640.
Definition
The Birkhoff orthogonality relation in a normed space X is calledε-symmetric (for some ε ∈ [0, 1)), if for any x , y ∈ X :
x⊥By =⇒ y⊥εBx .
⊥B is ε-symmetric for some ε ∈ [0, 1) if and only if:
x⊥By =⇒ ∃ z ∈ Lin {x , y} : y⊥Bz , ‖z − x‖ ≤ ε‖x‖.
The approximate symmetry of ⊥B does not imply that the norm comesfrom an inner product (even if dimX ≥ 3).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 15 / 28
Page 57
Sufficient conditions for approximate symmetry of ⊥B
A modulus of convexity of a normed space X , δX : [0, 2]→ [0, 1]:
δX (ε) := inf{
1−∥∥∥x + y
2
∥∥∥ : ‖x‖ ≤ 1, ‖y‖ ≤ 1, ‖x − y‖ ≥ ε}.
Theorem
If δX (1) > 0 and 1− 2δX (1) ≤ ε < 1, relation ⊥B is ε-symmetric.
Corollary
Suppose that for any ε ∈ [0, 1) the relation ⊥B is not ε-symmetric. Then
ε0(X ) := sup{ε ∈ [0, 2] : δX (ε) = 0} ≥ 1.
Moreover, if X is finite-dimensional, then R(X ) ≥ 1.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 16 / 28
Page 58
Sufficient conditions for approximate symmetry of ⊥B
A modulus of convexity of a normed space X , δX : [0, 2]→ [0, 1]:
δX (ε) := inf{
1−∥∥∥x + y
2
∥∥∥ : ‖x‖ ≤ 1, ‖y‖ ≤ 1, ‖x − y‖ ≥ ε}.
Theorem
If δX (1) > 0 and 1− 2δX (1) ≤ ε < 1, relation ⊥B is ε-symmetric.
Corollary
Suppose that for any ε ∈ [0, 1) the relation ⊥B is not ε-symmetric. Then
ε0(X ) := sup{ε ∈ [0, 2] : δX (ε) = 0} ≥ 1.
Moreover, if X is finite-dimensional, then R(X ) ≥ 1.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 16 / 28
Page 59
Sufficient conditions for approximate symmetry of ⊥B
A modulus of convexity of a normed space X , δX : [0, 2]→ [0, 1]:
δX (ε) := inf{
1−∥∥∥x + y
2
∥∥∥ : ‖x‖ ≤ 1, ‖y‖ ≤ 1, ‖x − y‖ ≥ ε}.
Theorem
If δX (1) > 0 and 1− 2δX (1) ≤ ε < 1, relation ⊥B is ε-symmetric.
Corollary
Suppose that for any ε ∈ [0, 1) the relation ⊥B is not ε-symmetric. Then
ε0(X ) := sup{ε ∈ [0, 2] : δX (ε) = 0} ≥ 1.
Moreover, if X is finite-dimensional, then R(X ) ≥ 1.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 16 / 28
Page 60
Sufficient conditions for approximate symmetry of ⊥B
A modulus of convexity of a normed space X , δX : [0, 2]→ [0, 1]:
δX (ε) := inf{
1−∥∥∥x + y
2
∥∥∥ : ‖x‖ ≤ 1, ‖y‖ ≤ 1, ‖x − y‖ ≥ ε}.
Theorem
If δX (1) > 0 and 1− 2δX (1) ≤ ε < 1, relation ⊥B is ε-symmetric.
Corollary
Suppose that for any ε ∈ [0, 1) the relation ⊥B is not ε-symmetric. Then
ε0(X ) := sup{ε ∈ [0, 2] : δX (ε) = 0} ≥ 1.
Moreover, if X is finite-dimensional, then R(X ) ≥ 1.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 16 / 28
Page 61
Sufficient conditions for approximate symmetry of ⊥B
Theorem
Let X be a real, uniformly convex normed space.Then, ⊥B is approximately-symmetric.
Theorem
Let X be a finite-dimensional real smooth normed space.Then, ⊥B is approximately-symmetric.
Theorem
Let X be a real uniformly convex and smooth Banach space. Then, theBirkhoff orthogonality relations in X , X ∗ and X ∗∗ are approximatelysymmetric (actually, with the same ε).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 17 / 28
Page 62
Sufficient conditions for approximate symmetry of ⊥B
Theorem
Let X be a real, uniformly convex normed space.Then, ⊥B is approximately-symmetric.
Theorem
Let X be a finite-dimensional real smooth normed space.Then, ⊥B is approximately-symmetric.
Theorem
Let X be a real uniformly convex and smooth Banach space. Then, theBirkhoff orthogonality relations in X , X ∗ and X ∗∗ are approximatelysymmetric (actually, with the same ε).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 17 / 28
Page 63
Sufficient conditions for approximate symmetry of ⊥B
Theorem
Let X be a real, uniformly convex normed space.Then, ⊥B is approximately-symmetric.
Theorem
Let X be a finite-dimensional real smooth normed space.Then, ⊥B is approximately-symmetric.
Theorem
Let X be a real uniformly convex and smooth Banach space. Then, theBirkhoff orthogonality relations in X , X ∗ and X ∗∗ are approximatelysymmetric (actually, with the same ε).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 17 / 28
Page 64
There are spaces in which the Birkhoff orthogonality is not approximatelysymmetric, i.e., for any ε ∈ [0, 1), ⊥B is not ε-symmetric.
Example
X = R2 with the maximum norm.
x
yz
y+λx
y + λz
Figure: x⊥By , y 6⊥Bx , y 6⊥Bz , y 6⊥εBx
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 18 / 28
Page 65
There are spaces in which the Birkhoff orthogonality is not approximatelysymmetric, i.e., for any ε ∈ [0, 1), ⊥B is not ε-symmetric.
Example
X = R2 with the maximum norm.
x
yz
y+λx
y + λz
Figure: x⊥By , y 6⊥Bx , y 6⊥Bz , y 6⊥εBx
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 18 / 28
Page 66
There are spaces in which the Birkhoff orthogonality is not approximatelysymmetric, i.e., for any ε ∈ [0, 1), ⊥B is not ε-symmetric.
Example
X = R2 with the maximum norm.
x
yz
y+λx
y + λz
Figure: x⊥By , y 6⊥Bx , y 6⊥Bz , y 6⊥εBx
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 18 / 28
Page 67
Geometrical properties connected with approximatesymmetry of B-orthogonality
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 19 / 28
Page 68
R(X ) := sup{‖x − y‖ : conv {x , y} ⊂ SX}.
We consider the following property of X :
x , y ∈ X , x 6= y , conv {x , y} ⊂ SX =⇒ X is smooth at x − y . (∗)
Examples
Each smooth or strictly convex space satisfies (∗).R2 with the supremum norm (which is neither strictly convex nor smooth)also satisfies (∗).X = R2 with the norm for which the unit ball is a symmetric polygon suchthat sides are not parallel to diagonals, the condition (∗) is satisfied.
Theorem
Let X be a two-dimensional strictly convex normed space and let Y be astrictly convex and smooth normed space. Then the space L (X ,Y )satisfies (∗).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 20 / 28
Page 69
R(X ) := sup{‖x − y‖ : conv {x , y} ⊂ SX}.
We consider the following property of X :
x , y ∈ X , x 6= y , conv {x , y} ⊂ SX =⇒ X is smooth at x − y . (∗)
Examples
Each smooth or strictly convex space satisfies (∗).R2 with the supremum norm (which is neither strictly convex nor smooth)also satisfies (∗).X = R2 with the norm for which the unit ball is a symmetric polygon suchthat sides are not parallel to diagonals, the condition (∗) is satisfied.
Theorem
Let X be a two-dimensional strictly convex normed space and let Y be astrictly convex and smooth normed space. Then the space L (X ,Y )satisfies (∗).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 20 / 28
Page 70
R(X ) := sup{‖x − y‖ : conv {x , y} ⊂ SX}.
We consider the following property of X :
x , y ∈ X , x 6= y , conv {x , y} ⊂ SX =⇒ X is smooth at x − y . (∗)
Examples
Each smooth or strictly convex space satisfies (∗).
R2 with the supremum norm (which is neither strictly convex nor smooth)also satisfies (∗).X = R2 with the norm for which the unit ball is a symmetric polygon suchthat sides are not parallel to diagonals, the condition (∗) is satisfied.
Theorem
Let X be a two-dimensional strictly convex normed space and let Y be astrictly convex and smooth normed space. Then the space L (X ,Y )satisfies (∗).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 20 / 28
Page 71
R(X ) := sup{‖x − y‖ : conv {x , y} ⊂ SX}.
We consider the following property of X :
x , y ∈ X , x 6= y , conv {x , y} ⊂ SX =⇒ X is smooth at x − y . (∗)
Examples
Each smooth or strictly convex space satisfies (∗).R2 with the supremum norm (which is neither strictly convex nor smooth)also satisfies (∗).
X = R2 with the norm for which the unit ball is a symmetric polygon suchthat sides are not parallel to diagonals, the condition (∗) is satisfied.
Theorem
Let X be a two-dimensional strictly convex normed space and let Y be astrictly convex and smooth normed space. Then the space L (X ,Y )satisfies (∗).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 20 / 28
Page 72
R(X ) := sup{‖x − y‖ : conv {x , y} ⊂ SX}.
We consider the following property of X :
x , y ∈ X , x 6= y , conv {x , y} ⊂ SX =⇒ X is smooth at x − y . (∗)
Examples
Each smooth or strictly convex space satisfies (∗).R2 with the supremum norm (which is neither strictly convex nor smooth)also satisfies (∗).X = R2 with the norm for which the unit ball is a symmetric polygon suchthat sides are not parallel to diagonals, the condition (∗) is satisfied.
Theorem
Let X be a two-dimensional strictly convex normed space and let Y be astrictly convex and smooth normed space. Then the space L (X ,Y )satisfies (∗).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 20 / 28
Page 73
R(X ) := sup{‖x − y‖ : conv {x , y} ⊂ SX}.
We consider the following property of X :
x , y ∈ X , x 6= y , conv {x , y} ⊂ SX =⇒ X is smooth at x − y . (∗)
Examples
Each smooth or strictly convex space satisfies (∗).R2 with the supremum norm (which is neither strictly convex nor smooth)also satisfies (∗).X = R2 with the norm for which the unit ball is a symmetric polygon suchthat sides are not parallel to diagonals, the condition (∗) is satisfied.
Theorem
Let X be a two-dimensional strictly convex normed space and let Y be astrictly convex and smooth normed space. Then the space L (X ,Y )satisfies (∗).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 20 / 28
Page 74
Theorem
Let X be a real normed space satisfying (∗) and let ε ∈ (0, 1). If theorthogonality relation ⊥B in X is ε-symmetric, then R(X ) ≤ 2ε.
Corollary
If X is a real normed space satisfying (∗) and R(X ) = 2, then the Birkhofforthogonality in X is not approximately symmetric.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 21 / 28
Page 75
Theorem
Let X be a real normed space satisfying (∗) and let ε ∈ (0, 1). If theorthogonality relation ⊥B in X is ε-symmetric, then R(X ) ≤ 2ε.
Corollary
If X is a real normed space satisfying (∗) and R(X ) = 2, then the Birkhofforthogonality in X is not approximately symmetric.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 21 / 28
Page 76
Symmetry constant S(X )
Definition
S(X ) := inf{ε ∈ [0, 1] : ∀ x , y ∈ X x⊥By ⇒ y⊥εBx}.
S(X ) ∈ [0, 1]S(X ) = 0 means that ⊥B is symmetric.S(X ) = 1 means that ⊥B is not approximately symmetric.
S(X ) = sup{S(X0) : X0 is a two-dimensional subspace of X}.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 22 / 28
Page 77
Symmetry constant S(X )
Definition
S(X ) := inf{ε ∈ [0, 1] : ∀ x , y ∈ X x⊥By ⇒ y⊥εBx}.
S(X ) ∈ [0, 1]S(X ) = 0 means that ⊥B is symmetric.S(X ) = 1 means that ⊥B is not approximately symmetric.
S(X ) = sup{S(X0) : X0 is a two-dimensional subspace of X}.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 22 / 28
Page 78
Symmetry constant S(X )
Definition
S(X ) := inf{ε ∈ [0, 1] : ∀ x , y ∈ X x⊥By ⇒ y⊥εBx}.
S(X ) ∈ [0, 1]
S(X ) = 0 means that ⊥B is symmetric.S(X ) = 1 means that ⊥B is not approximately symmetric.
S(X ) = sup{S(X0) : X0 is a two-dimensional subspace of X}.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 22 / 28
Page 79
Symmetry constant S(X )
Definition
S(X ) := inf{ε ∈ [0, 1] : ∀ x , y ∈ X x⊥By ⇒ y⊥εBx}.
S(X ) ∈ [0, 1]S(X ) = 0 means that ⊥B is symmetric.
S(X ) = 1 means that ⊥B is not approximately symmetric.
S(X ) = sup{S(X0) : X0 is a two-dimensional subspace of X}.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 22 / 28
Page 80
Symmetry constant S(X )
Definition
S(X ) := inf{ε ∈ [0, 1] : ∀ x , y ∈ X x⊥By ⇒ y⊥εBx}.
S(X ) ∈ [0, 1]S(X ) = 0 means that ⊥B is symmetric.S(X ) = 1 means that ⊥B is not approximately symmetric.
S(X ) = sup{S(X0) : X0 is a two-dimensional subspace of X}.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 22 / 28
Page 81
Symmetry constant S(X )
Definition
S(X ) := inf{ε ∈ [0, 1] : ∀ x , y ∈ X x⊥By ⇒ y⊥εBx}.
S(X ) ∈ [0, 1]S(X ) = 0 means that ⊥B is symmetric.S(X ) = 1 means that ⊥B is not approximately symmetric.
S(X ) = sup{S(X0) : X0 is a two-dimensional subspace of X}.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 22 / 28
Page 82
If X is uniformly convex, then S(X ) < 1 (the reverse is not true).
If X satisfies (∗), then1
2R(X ) ≤ S(X ).
If X is a real uniformly convex and smooth Banach space, then
S(X ) = S(X ∗) = S(X ∗∗) < 1.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 23 / 28
Page 83
If X is uniformly convex, then S(X ) < 1 (the reverse is not true).
If X satisfies (∗), then1
2R(X ) ≤ S(X ).
If X is a real uniformly convex and smooth Banach space, then
S(X ) = S(X ∗) = S(X ∗∗) < 1.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 23 / 28
Page 84
If X is uniformly convex, then S(X ) < 1 (the reverse is not true).
If X satisfies (∗), then1
2R(X ) ≤ S(X ).
If X is a real uniformly convex and smooth Banach space, then
S(X ) = S(X ∗) = S(X ∗∗) < 1.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 23 / 28
Page 85
Example
Let X = R2 with an l1-l∞ norm.
For δ > 0 let Y = Yδ = R2 with the norm such that the unit sphere is ahexagon with vertices at
y1 = (1, 0), y2 = (0, 1), y3 = (−1− δ, 1− δ), y4 = (−1, 0),
y5 = (0,−1), y6 = (1 + δ,−1 + δ).
x1 = y1
x2 = y2
x3
y3
x4 = y4
x5 = y5 x6
y6
Figure: Unit spheres in X and Y
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 24 / 28
Page 86
Example
Let X = R2 with an l1-l∞ norm.For δ > 0 let Y = Yδ = R2 with the norm such that the unit sphere is ahexagon with vertices at
y1 = (1, 0), y2 = (0, 1), y3 = (−1− δ, 1− δ), y4 = (−1, 0),
y5 = (0,−1), y6 = (1 + δ,−1 + δ).
x1 = y1
x2 = y2
x3
y3
x4 = y4
x5 = y5 x6
y6
Figure: Unit spheres in X and Y
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 24 / 28
Page 87
Example
Let X = R2 with an l1-l∞ norm.For δ > 0 let Y = Yδ = R2 with the norm such that the unit sphere is ahexagon with vertices at
y1 = (1, 0), y2 = (0, 1), y3 = (−1− δ, 1− δ), y4 = (−1, 0),
y5 = (0,−1), y6 = (1 + δ,−1 + δ).
x1 = y1
x2 = y2
x3
y3
x4 = y4
x5 = y5 x6
y6
Figure: Unit spheres in X and Y
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 24 / 28
Page 88
It can be checked that the Banach-Mazur distance d(X ,Y ) can bearbitrarily close to 1; namely:
d(X ,Y ) ≤ 1 + δ
1− δ.
The space X is a Radon plane, therefore S(X ) = 0.
No matter how small is δ > 0, the space Y satisfies (∗), whence
S(Y ) ≥ 1
2R(Y ) >
1
2.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 25 / 28
Page 89
It can be checked that the Banach-Mazur distance d(X ,Y ) can bearbitrarily close to 1; namely:
d(X ,Y ) ≤ 1 + δ
1− δ.
The space X is a Radon plane, therefore S(X ) = 0.
No matter how small is δ > 0, the space Y satisfies (∗), whence
S(Y ) ≥ 1
2R(Y ) >
1
2.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 25 / 28
Page 90
It can be checked that the Banach-Mazur distance d(X ,Y ) can bearbitrarily close to 1; namely:
d(X ,Y ) ≤ 1 + δ
1− δ.
The space X is a Radon plane, therefore S(X ) = 0.
No matter how small is δ > 0, the space Y satisfies (∗)
, whence
S(Y ) ≥ 1
2R(Y ) >
1
2.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 25 / 28
Page 91
It can be checked that the Banach-Mazur distance d(X ,Y ) can bearbitrarily close to 1; namely:
d(X ,Y ) ≤ 1 + δ
1− δ.
The space X is a Radon plane, therefore S(X ) = 0.
No matter how small is δ > 0, the space Y satisfies (∗), whence
S(Y ) ≥ 1
2R(Y ) >
1
2.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 25 / 28
Page 92
S(lpn ) (p > 1)
The Banach-Mazur distance between lpn and l2n is equal to:
d := d(lpn , l2n ) = n
∣∣∣ 1p− 1
2
∣∣∣.
We were able to estimate that for p > 1, sufficiently close to 2, we have
S(lpn ) ≤ max
{(2p −
(1 +
1
d2
)p) 1p
,
(2q −
(1 +
1
d2
)q) 1q
}
(with q such that 1p + 1
q = 1).If p → 2, then q → 2 and d → 1.Therefore
limp→2
S(lpn ) = 0 = S(l2n ).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 26 / 28
Page 93
S(lpn ) (p > 1)
The Banach-Mazur distance between lpn and l2n is equal to:
d := d(lpn , l2n ) = n
∣∣∣ 1p− 1
2
∣∣∣.
We were able to estimate that for p > 1, sufficiently close to 2, we have
S(lpn ) ≤ max
{(2p −
(1 +
1
d2
)p) 1p
,
(2q −
(1 +
1
d2
)q) 1q
}
(with q such that 1p + 1
q = 1).If p → 2, then q → 2 and d → 1.Therefore
limp→2
S(lpn ) = 0 = S(l2n ).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 26 / 28
Page 94
S(lpn ) (p > 1)
The Banach-Mazur distance between lpn and l2n is equal to:
d := d(lpn , l2n ) = n
∣∣∣ 1p− 1
2
∣∣∣.
We were able to estimate that for p > 1, sufficiently close to 2, we have
S(lpn ) ≤ max
{(2p −
(1 +
1
d2
)p) 1p
,
(2q −
(1 +
1
d2
)q) 1q
}
(with q such that 1p + 1
q = 1).If p → 2, then q → 2 and d → 1.Therefore
limp→2
S(lpn ) = 0 = S(l2n ).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 26 / 28
Page 95
S(lpn ) (p > 1)
The Banach-Mazur distance between lpn and l2n is equal to:
d := d(lpn , l2n ) = n
∣∣∣ 1p− 1
2
∣∣∣.
We were able to estimate that for p > 1, sufficiently close to 2, we have
S(lpn ) ≤ max
{(2p −
(1 +
1
d2
)p) 1p
,
(2q −
(1 +
1
d2
)q) 1q
}
(with q such that 1p + 1
q = 1).
If p → 2, then q → 2 and d → 1.Therefore
limp→2
S(lpn ) = 0 = S(l2n ).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 26 / 28
Page 96
S(lpn ) (p > 1)
The Banach-Mazur distance between lpn and l2n is equal to:
d := d(lpn , l2n ) = n
∣∣∣ 1p− 1
2
∣∣∣.
We were able to estimate that for p > 1, sufficiently close to 2, we have
S(lpn ) ≤ max
{(2p −
(1 +
1
d2
)p) 1p
,
(2q −
(1 +
1
d2
)q) 1q
}
(with q such that 1p + 1
q = 1).If p → 2, then q → 2 and d → 1.
Thereforelimp→2
S(lpn ) = 0 = S(l2n ).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 26 / 28
Page 97
S(lpn ) (p > 1)
The Banach-Mazur distance between lpn and l2n is equal to:
d := d(lpn , l2n ) = n
∣∣∣ 1p− 1
2
∣∣∣.
We were able to estimate that for p > 1, sufficiently close to 2, we have
S(lpn ) ≤ max
{(2p −
(1 +
1
d2
)p) 1p
,
(2q −
(1 +
1
d2
)q) 1q
}
(with q such that 1p + 1
q = 1).If p → 2, then q → 2 and d → 1.Therefore
limp→2
S(lpn ) = 0 = S(l2n ).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 26 / 28
Page 98
Thank you!
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 27 / 28
Page 99
Bibliography
J. Alonso, H. Martini, S. Wu, On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces, Aequationes
Math. 83, No. 1-2 (2012), 153-189.
R. Bhatia, P. Semrl, Orthogonality of matrices and some distance problems, Linear Algebra Appl. 287 (1999), 77-85.
G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J., 1 (1935), 169–172.
J. Chmielinski, On an ε-Birkhoff orthogonality, J. Inequal. Pure and Appl. Math., 6(3) (2005), Art. 79.
J. Chmielinski, T. Stypu la, P. Wojcik, Approximate orthogonality in normed spaces and its applications, Linear Algebra
Appl. 531 (2017), 305–317.
J. Chmielinski, P. Wojcik, Approximate symmetry of the Birkhoff orthogonality, J. Math. Anal. Appl. 461 (2018),
625–640.
R.C. James, Orthogonality in normed linear linear spaces, Duke Math. J. 12 (1945), 291–301.
R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc., 61 (1947), 265–292.
R.C. James, Inner products in normed linear spaces, Bull. Am. Math. Soc. 53 (1947), 559-566.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 28 / 28