IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
On isometries on some Banach spaces– something old, something new,
something borrowed, something blue,
Part I
Dijana Ilisevic
University of Zagreb, Croatia
Recent Trends in Operator Theory and ApplicationsMemphis, TN, USA, May 3–5, 2018
Recent work of D.I. has been fully supported by the Croatian Science Foundation under the project
IP-2016-06-1046.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Something old, something new,
something borrowed, something blue
is referred to the collection of items that
helps to guarantee fertility and prosperity.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Isometries
Isometries are maps between metric spaces which preserve distancebetween elements.
Definition
Let (X , | · |) and (Y, ‖ · ‖) be two normed spaces over the samefield. A linear map ϕ : X → Y is called a linear isometry if
‖ϕ(x)‖ = |x |, x ∈ X .
We shall be interested in surjective linear isometries on Banachspaces.
One of the main problems is to give explicit description ofisometries on a particular space.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Isometries
Isometries are maps between metric spaces which preserve distancebetween elements.
Definition
Let (X , | · |) and (Y, ‖ · ‖) be two normed spaces over the samefield. A linear map ϕ : X → Y is called a linear isometry if
‖ϕ(x)‖ = |x |, x ∈ X .
We shall be interested in surjective linear isometries on Banachspaces.
One of the main problems is to give explicit description ofisometries on a particular space.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Books
Richard J. Fleming, James E. Jamison,
Isometries on Banach spaces: function spaces,Chapman & Hall/CRC, 2003 (208 pp.)
Isometries on Banach spaces: vector-valued function spaces,Chapman & Hall/CRC, 2008 (248 pp.)
This talk is dedicated to the memory of Professor James Jamison.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Trivial isometries
Trivial isometries are isometries of the form λI for some λ ∈ T,where T = λ ∈ F : |λ| = 1.
The spectrum of a surjective linear isometry is contained in T.
For any Banach space X (real or complex) there is a norm ‖ · ‖ onX , equivalent to the original one, such that (X , ‖ · ‖) has onlytrivial isometries (K. Jarosz, 1988).
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Trivial isometries
Trivial isometries are isometries of the form λI for some λ ∈ T,where T = λ ∈ F : |λ| = 1.
The spectrum of a surjective linear isometry is contained in T.
For any Banach space X (real or complex) there is a norm ‖ · ‖ onX , equivalent to the original one, such that (X , ‖ · ‖) has onlytrivial isometries (K. Jarosz, 1988).
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Trivial isometries
Trivial isometries are isometries of the form λI for some λ ∈ T,where T = λ ∈ F : |λ| = 1.
The spectrum of a surjective linear isometry is contained in T.
For any Banach space X (real or complex) there is a norm ‖ · ‖ onX , equivalent to the original one, such that (X , ‖ · ‖) has onlytrivial isometries (K. Jarosz, 1988).
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Norms induced by the inner product
Let V be a finite dimensional vector space over F ∈ R,C,equipped with the norm ‖ · ‖ induced by the inner product
〈x , y〉 = tr (xy∗) = y∗x
(Frobenius norm).Then U is a linear isometry on (V, ‖ · ‖) if and only if the followingholds.
If F = C: U is a unitary operator on V, that is,
U∗U = UU∗ = I .
If F = R: U is an orthogonal operator on V, that is,
UtU = UUt = I .
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Spectral norm
Theorem (I. Schur, 1925)
Linear isometries of Mn(C) equipped with the spectral norm(operator norm) have one of the following forms:
X 7→ UXV or X 7→ UX tV ,
where U,V ∈ Mn(C) are unitaries.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Unitarily invariant norms on Mn(F)
[C.K. Li, N.K. Tsing, 1990]
Let G be the group of all linear operators of the form X 7→ UXVfor some fixed unitary (orthogonal) U,V ∈ Mn(F).
A norm ‖ · ‖ on Mn(F) is called a unitarily invariant norm if‖g(X )‖ = ‖X‖ for all g ∈ G , X ∈ Mn(F).
If ‖ · ‖ is a unitarily invariant norm (which is not a multiple of theFrobenius norm) on Mn(F) 6= M4(R) then its isometry group is〈G , τ〉, where τ : Mn(F)→ Mn(F) is the transposition operator.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Unitarily invariant norms on M4(F)
In the case of M4(R) the isometry group is 〈G , τ〉 or 〈G , τ, α〉,with α : M4(R)→ M4(R) defined by
α(X ) = (X + B1XC1 + B2XC2 + B3XC3)/2, where
B1 =
(1 00 1
)⊗(
0 −11 0
), C1 =
(1 00 −1
)⊗(
0 1−1 0
),
B2 =
(0 −11 0
)⊗(
1 00 −1
), C2 =
(0 1−1 0
)⊗(
1 00 1
),
B3 =
(0 −11 0
)⊗(
0 11 0
), C3 =
(0 11 0
)⊗(
0 1−1 0
).
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Unitary congruence invariant norms on Sn(C)
[C.K. Li, N.K. Tsing, 1990-1991]
Let G be the group of all linear operators of the form X 7→ UtXUfor some fixed unitary (orthogonal) U ∈ Mn(F).
A norm ‖ · ‖ on V ∈ Sn(C),Kn(F) is called a unitarycongruence invariant norm if ‖g(X )‖ = ‖X‖ for all g ∈ G ,X ∈ V .
If ‖ · ‖ is a unitary congruence invariant norm on Sn(C), which isnot a multiple of the Frobenius norm, then its isometry group is G .
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Unitary congruence invariant norms on Kn(F)
If ‖ · ‖ is a unitary congruence invariant norm on Kn(C), which isnot a multiple of the Frobenius norm, then its isometry group is Gif n 6= 4, and 〈G , γ〉 if n = 4, where γ(X ) is obtained from X byinterchanging its (1, 4) and (2, 3) entries, and interchanging its(4, 1) and (3, 2) entries accordingly.
If ‖ · ‖ is a unitary congruence invariant norm on Kn(R), which isnot a multiple of the Frobenius norm, then its isometry group is〈G , τ〉 if n 6= 4, and 〈G , τ, γ〉 if n = 4.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Surjective linear isometries of C0(Ω)
Let C0(Ω) be the algebra of all continuous complex-valuedfunctions on a locally compact Hausdorff space Ω, vanishing atinfinity.
Theorem (Banach–Stone)
Let T : C0(Ω1)→ C0(Ω2) be a surjective linear isometry.Then there exist a homeomorphism ϕ : Ω2 → Ω1 and a continuousunimodular function u : Ω2 → C such that
T (f )(ω) = u(ω)f(ϕ(ω)
), f ∈ C0(Ω1), ω ∈ Ω2.
The first (Banach’s) version of this theorem (1932): for real-valuedfunctions on compact metric spaces.
Stone (1937): for real-valued functions on compact Hausdorff spaces.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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Surjective linear isometries of C0(Ω)
Let C0(Ω) be the algebra of all continuous complex-valuedfunctions on a locally compact Hausdorff space Ω, vanishing atinfinity.
Theorem (Banach–Stone)
Let T : C0(Ω1)→ C0(Ω2) be a surjective linear isometry.Then there exist a homeomorphism ϕ : Ω2 → Ω1 and a continuousunimodular function u : Ω2 → C such that
T (f )(ω) = u(ω)f(ϕ(ω)
), f ∈ C0(Ω1), ω ∈ Ω2.
The first (Banach’s) version of this theorem (1932): for real-valuedfunctions on compact metric spaces.
Stone (1937): for real-valued functions on compact Hausdorff spaces.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
C ∗-algebras
A C ∗-algebra is a complex Banach ∗-algebra (A, ‖ . ‖) such that‖a∗a‖ = ‖a‖2 for all a ∈ A.
Example
C = complex numbers,
B(H) = all bounded linear operators on a complex Hilbertspace H,
K(H) = all compact operators on a complex Hilbert space H,
C (Ω) = all continuous complex-valued functions on acompact Hausdorff space Ω,
C0(Ω) = all continuous complex-valued functions on a locallycompact Hausdorff space Ω, vanishing at infinity.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Isometries of C*-algebras
Theorem (R. Kadison, 1951)
Let A and B be unital C*-algebras and T : A → B a surjectivelinear isometry. Then T = UJ, where J : A → B is a Jordan∗-isomorphism (that is, a linear map satisfying J(a2) = J(a)2 andJ(a∗) = J(a)∗ for every a ∈ A) and a unitary element U ∈ B.
Theorem (A. Paterson, A. Sinclair, 1972)
Let A and B be C*-algebras and T : A → B a surjective linearisometry. Then T = UJ, where J : A → B is a Jordan∗-isomorphism, and U on B is unitary such that there exists V onB satisfying aU(b) = V (a)b for all a, b ∈ B.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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Isometries of C*-algebras
Theorem (R. Kadison, 1951)
Let A and B be unital C*-algebras and T : A → B a surjectivelinear isometry. Then T = UJ, where J : A → B is a Jordan∗-isomorphism (that is, a linear map satisfying J(a2) = J(a)2 andJ(a∗) = J(a)∗ for every a ∈ A) and a unitary element U ∈ B.
Theorem (A. Paterson, A. Sinclair, 1972)
Let A and B be C*-algebras and T : A → B a surjective linearisometry. Then T = UJ, where J : A → B is a Jordan∗-isomorphism, and U on B is unitary such that there exists V onB satisfying aU(b) = V (a)b for all a, b ∈ B.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Isometries of B(H)
Let B(H) be the algebra of all bounded linear operators on acomplex Hilbert space H. Throughout we fix an orthonormal basiseλ : λ ∈ Λ of H.
Let T ∈ B(H). If S ∈ B(H) is such that 〈Teλ, eµ〉 = 〈Seµ, eλ〉 forall λ, µ ∈ Λ, then S is called the transpose of T associated to thebasis eλ : λ ∈ Λ and it is denoted by T t .
Theorem
Let T : B(H)→ B(H) be a surjective linear isometry. Then thereexist unitary U,V ∈ B(H) such that T has one of the followingforms:
X 7→ UXV or X 7→ UX tV .
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
JB*-triples
A JB*-triple is a complex Banach space A together with acontinuous triple product · · · : A×A×A → A such that
(i) xyz is linear in x and z and conjugate linear in y ;(ii) xyz = zyx;(iii) for any x ∈ A, the operator δ(x) : A → A defined by
δ(x)y = xxy is hermitian with nonnegative spectrum;(iv) δ(x)abc = δ(x)a, b, c − a, δ(x)b, c+ a, b, δ(x)c;(v) for every x ∈ A, ‖xxx‖ = ‖x‖3.
Example
complex Hilbert spaces: xyz = 12 (〈x , y〉z + 〈z , y〉x)
C*-algebras, S(H), A(H): xyz = 12 (xy∗z + zy∗x), where
S(H) = T ∈ B(H) : T t = T symmetric operators,
A(H) = T ∈ B(H) : T t = −T antisymmetric operators.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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Isometries on JB*-triples
Theorem (W. Kaup, 1983)
Let A be a JB*-triple. Then every surjective linear isometryT : A → A satisfies
T (xyz) = T (x)T (y)T (z), x , y , z ∈ A.
In particular, if A is a C*-algebra then
T (xy∗x) = T (x)T (y)∗T (x), x , y ∈ A.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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Isometries on JB*-triples
Theorem (W. Kaup, 1983)
Let A be a JB*-triple. Then every surjective linear isometryT : A → A satisfies
T (xyz) = T (x)T (y)T (z), x , y , z ∈ A.
In particular, if A is a C*-algebra then
T (xy∗x) = T (x)T (y)∗T (x), x , y ∈ A.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Isometries on S(H) and A(H)
Every surjective linear isometry T : A → A, where A is S(H) orA(H), satisfies
T (XY ∗X ) = T (X )T (Y )∗T (X )
for all X ,Y ∈ A.
The following theorem gives an explicit formula for T .
Theorem (A. Fosner and D. I., 2011)
Let A be S(H) or A(H) and let T : A → A be a surjective linearisometry. Then there exists a unitary U ∈ B(H) such that T hasthe form X 7→ UXUt .
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Isometries on S(H) and A(H)
Every surjective linear isometry T : A → A, where A is S(H) orA(H), satisfies
T (XY ∗X ) = T (X )T (Y )∗T (X )
for all X ,Y ∈ A.
The following theorem gives an explicit formula for T .
Theorem (A. Fosner and D. I., 2011)
Let A be S(H) or A(H) and let T : A → A be a surjective linearisometry. Then there exists a unitary U ∈ B(H) such that T hasthe form X 7→ UXUt .
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Minimal norm ideals in B(H)
A minimal norm ideal (I, ν) consists of a two-sided proper idealI in B(H) together with a norm ν on I satisfying the following:
the set of all finite rank operators on H is dense in I,
ν(X ) = ‖X‖ for every rank one operator X ,
ν(UXV ) = ν(X ) for every X ∈ I and all unitaryU,V ∈ B(H).
Theorem (A. Sourour, 1981)
If I is different from the Hilbert-Schmidt class then everysurjective linear isometry on I has the form X 7→ UXV orX 7→ UX tV for some unitary U,V ∈ B(H).
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Hermitian operators
Definition
Let X be a complex Banach space. A bounded linear operatorT : X → X is said to be hermitian if e iϕT is an isometry for allϕ ∈ R.
Example
C 1[0, 1], the space of continuously differentiable complex-valuedfunctions on [0, 1] with ‖f ‖ = ‖f ‖∞ + ‖f ′‖∞, admits only trivialhermitian operators, that is, real multiples of I (E. Berkson,A. Sourour, 1974).
Example
Hermitian operators on a C*-algebra A have the form x 7→ ax + xbfor some self-adjoint a, b ∈ M(A).
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Hermitian operators
Definition
Let X be a complex Banach space. A bounded linear operatorT : X → X is said to be hermitian if e iϕT is an isometry for allϕ ∈ R.
Example
C 1[0, 1], the space of continuously differentiable complex-valuedfunctions on [0, 1] with ‖f ‖ = ‖f ‖∞ + ‖f ′‖∞, admits only trivialhermitian operators, that is, real multiples of I (E. Berkson,A. Sourour, 1974).
Example
Hermitian operators on a C*-algebra A have the form x 7→ ax + xbfor some self-adjoint a, b ∈ M(A).
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Hermitian projections
By a projection on a complex Banach space we mean a linearoperator P such that P2 = P.
Theorem (J. Jamison, 2007)
A projection P on a complex Banach space is a hermitianprojection if and only if P + λ(I − P) is an isometry for all λ ∈ T,where T = λ ∈ C : |λ| = 1.
Example
C 1[0, 1] admits only trivial hermitian projections (0 and I ).
Example
Every orthogonal projection on a complex Hilbert space ishermitian.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Hermitian projections
By a projection on a complex Banach space we mean a linearoperator P such that P2 = P.
Theorem (J. Jamison, 2007)
A projection P on a complex Banach space is a hermitianprojection if and only if P + λ(I − P) is an isometry for all λ ∈ T,where T = λ ∈ C : |λ| = 1.
Example
C 1[0, 1] admits only trivial hermitian projections (0 and I ).
Example
Every orthogonal projection on a complex Hilbert space ishermitian.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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Hermitian projections on some operator spaces
Theorem (L.L. Stacho and B. Zalar, 2004)
(i) Let P : B(H)→ B(H) be a hermitian projection. Then P hasthe form X 7→ QX or X 7→ XQ for some Q ∈ B(H) such thatQ = Q∗ = Q2.
(ii) Let P : S(H)→ S(H) be a hermitian projection. Then eitherP = 0 or P = I .
(iii) Let P : A(H)→ A(H) be a hermitian projection. Then P orI − P has the form X 7→ QX + XQt with Q = x ⊗ x for someunit vector x ∈ H.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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Hermitian projections on C*-algebras
Theorem (M. Fosner and D. I., 2005)
Let A be a C*-algebra and let P : A → A be a hermitianprojection. Then there exist a ∗-ideal I of A andp = p∗ = p2 ∈ M(I⊥ ⊕ I⊥⊥) such that P(x) = px for all x ∈ I⊥and P(x) = xp for all x ∈ I⊥⊥.
Corollary
Let Ω be a locally compact Hausdorff space. ThenP : C0(Ω)→ C0(Ω) is a hermitian projection if and only ifPf = 1Y f , where 1Y f is the indicator function on a propercomponent Y of Ω.In particular, if Ω is connected then C0(Ω) admits only trivialhermitian projections.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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Hermitian projections on minimal norm ideals
Corollary
Let A be K (H) or B(H) and let P : A → A be a hermitianprojection. Then there exists p = p∗ = p2 ∈ B(H) such that P hasthe form x 7→ px or x 7→ xp.
Theorem (J. Jamison, 2007)
Let I be a minimal norm ideal in B(H), different from theHilbert-Schmidt class, and let P : I → I be a hermitian projection.Then P has the form X 7→ QX or X 7→ XQ for someQ = Q∗ = Q2 ∈ B(H).
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Hermitian projectionsGeneralized bicircular projectionsGeneralized n-circular projections
Hermitian projections on minimal norm ideals
Corollary
Let A be K (H) or B(H) and let P : A → A be a hermitianprojection. Then there exists p = p∗ = p2 ∈ B(H) such that P hasthe form x 7→ px or x 7→ xp.
Theorem (J. Jamison, 2007)
Let I be a minimal norm ideal in B(H), different from theHilbert-Schmidt class, and let P : I → I be a hermitian projection.Then P has the form X 7→ QX or X 7→ XQ for someQ = Q∗ = Q2 ∈ B(H).
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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A generalization of hermitian projections
Recall that a projection P on X is a hermitian projection if andonly if the map
P + λ(I − P) is an isometry for all λ ∈ T.
These projections are also known as bicircular projections.
We can also study projections P such that
P + λ(I − P) is an isometry for some λ ∈ T \ 1.
These projections are also known as generalized bicircularprojections (GBPs).
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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Generalized bicircular projections on Sn(C)
Let A be Sn(C) or Kn(C). A norm ‖ · ‖ on A is said to be aunitary congruence invariant norm if
‖UXUt‖ = ‖X‖
for all unitary U ∈ Mn(C) and all X ∈ A.
Theorem (M. Fosner, D. I. and C.K. Li, 2007)
Let ‖ · ‖ be a unitary congruence invariant norm on Sn(C), which isnot a multiple of the Frobenius norm. Suppose P : Sn(C)→ Sn(C)is a nontrivial projection and λ ∈ T \ 1. Then P + λ(I − P) is anisometry of (Sn(C), ‖ · ‖) if and only if λ = −1 and there existsQ = Q∗ = Q2 ∈ Mn(C) such that P or I − P has the formX 7→ QXQt + (I − Q)X (I − Qt).
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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Generalized bicircular projections on Sn(C)
Let A be Sn(C) or Kn(C). A norm ‖ · ‖ on A is said to be aunitary congruence invariant norm if
‖UXUt‖ = ‖X‖
for all unitary U ∈ Mn(C) and all X ∈ A.
Theorem (M. Fosner, D. I. and C.K. Li, 2007)
Let ‖ · ‖ be a unitary congruence invariant norm on Sn(C), which isnot a multiple of the Frobenius norm. Suppose P : Sn(C)→ Sn(C)is a nontrivial projection and λ ∈ T \ 1. Then P + λ(I − P) is anisometry of (Sn(C), ‖ · ‖) if and only if λ = −1 and there existsQ = Q∗ = Q2 ∈ Mn(C) such that P or I − P has the formX 7→ QXQt + (I − Q)X (I − Qt).
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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Generalized bicircular projections on Kn(C)
Theorem (M. Fosner, D. I. and C.K. Li, 2007)
Let ‖ · ‖ be a unitary congruence invariant norm on Kn(C), which isnot a multiple of the Frobenius norm. Suppose P : Kn(C)→ Kn(C)is a nontrivial projection and λ ∈ T \ 1. Then P + λ(I − P) is anisometry of (Kn(C), ‖ · ‖) if and only if one of the following holds.
(i) There exists Q = vv∗ for a unit vector v ∈ Cn such that P orI − P has the form X 7→ QX + XQt .
(ii) λ = −1, K = G and there exists Q = Q∗ = Q2 ∈ Mn(C) suchthat P or I −P has the form X 7→ QXQt + (I −Q)X (I −Qt).
(iii) (λ, n) = (−1, 4), ψ ∈ K, and there is a unitary U ∈ M4(C),satisfying ψ(UtXU) = Uψ(X )U∗ for all X ∈ K4(C), such thatP or I − P has the formX 7→ (X + ψ(UtXU))/2 = (X + Uψ(X )U∗)/2.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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Generalized bicircular projections on minimal norm ideals
Theorem (F. Botelho and J. Jamison, 2008)
Let I be a minimal norm ideal in B(H), different from theHilbert-Schmidt class, and let P : A → A be a projection. ThenP + λ(I − P) is an isometry for some λ ∈ T \ 1 if and only if oneof the following holds:
(i) P has the form X 7→ QX or X 7→ XQ for someQ = Q∗ = Q2 ∈ B(H),
(ii) λ = −1 and P has one of the following forms:
X 7→ 12 (X + UXV ) for some unitary U,V ∈ B(H) such that
U2 = µI , V 2 = µI for some µ ∈ C, |µ| = 1,X 7→ 1
2 (X + UX tV ) for some unitary U,V ∈ B(H) such thatV = ±(U t)∗.
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Generalized bicircular projections on arbitrary complex Banach spaces
Theorem (P.-K. Lin, 2008)
Let X be a complex Banach space and let P : X → X be aprojection. Then P + λ(I − P) is an isometry for some λ ∈ T \ 1if and only if one of the following holds:
(i) P is hermitian,
(ii) λ = e2πin for some integer n ≥ 2.
Furthermore, if n is any integer such that n ≥ 2, then for λ = e2πin
there is a complex Banach space X and a nontrivial projection Pon X such that P + λ(I − P) is an isometry.
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Generalized bicircular projections on JB*-triples
Theorem (D. I., 2010)
Let A be a JB*-triple and let P : A → A be a projection. ThenP + λ(I − P) is an isometry for some λ ∈ T \ 1 if and only if oneof the following holds:
(i) P is hermitian,
(ii) λ = −1 and P = 12 (I + T ) for some linear isometry
T : A → A satisfying T 2 = I .
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Corollary
Let A = B(H) or A = K (H), and let P : A → A be anonhermitian projection. Then P + λ(I − P) is an isometry forsome λ ∈ T \ 1 if and only if λ = −1 and P has one of thefollowing forms:
X 7→ 12 (X + UXV ) for unitary U,V ∈ B(H) such that
U2 = µI , V 2 = µI for some µ ∈ C, |µ| = 1,
X 7→ 12 (X + UX tV ) for unitary U,V ∈ B(H) such that
V = ±(Ut)∗.
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Theorem (F. Botelho, 2008)
Let Ω be a connected compact Hausdorff space and letP : C (Ω)→ C (Ω) be a nontrivial projection.Then P + λ(I − P) is an isometry for some λ ∈ T \ 1 if and onlyif λ = −1 and there exist a homeomorphism ϕ : Ω→ Ω satisfyingϕ2 = I and a continuous unimodular function u : Ω→ C satisfyingu(ϕ(ω)) = u(ω) for every ω ∈ Ω, such that
P(f )(ω) =1
2
(f (ω) + u(ω)f
(ϕ(ω)
)), f ∈ C0(Ω), ω ∈ Ω.
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Theorem (D. I., 2010)
Let Ω be a locally compact Hausdorff space and letP : C0(Ω)→ C0(Ω) be a projection. Then P + λ(I − P) is anisometry for some λ ∈ T \ 1 if and only if one of the followingholds.
(i) P is hermitian,
(ii) λ = −1 and there exist a homeomorphism ϕ : Ω→ Ωsatisfying ϕ2 = I and a continuous unimodular functionu : Ω→ C satisfying u(ϕ(ω)) = u(ω) for every ω ∈ Ω, suchthat
P(f )(ω) =1
2
(f (ω) + u(ω)f
(ϕ(ω)
)), f ∈ C0(Ω), ω ∈ Ω.
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Generalized bicircular projections on S(H) and A(H)
Corollary (A. Fosner and D. I., 2011)
Let P : S(H)→ S(H) be a nontrivial projection and λ ∈ T \ 1. ThenP + λ(I − P) is an isometry if and only if λ = −1 and there existsQ = Q∗ = Q2 ∈ B(H) such that P or I − P has the formX 7→ QXQt + (I − Q)X (I − Qt).
Corollary (A. Fosner and D. I., 2011)
Let P : A(H)→ A(H) be a nontrivial projection and λ ∈ T \ 1. ThenP + λ(I − P) is an isometry if and only if one of the following holds:
(i) P or I − P has the form X 7→ QX + XQt , where Q = x ⊗ x forsome norm one x ∈ H,
(ii) λ = −1 and there exists Q = Q∗ = Q2 ∈ B(H) such that P orI − P has the form X 7→ QXQt + (I − Q)X (I − Qt).
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Generalized bicircular projectionsand the spectrum of the corresponding isometry
If P is a projection such that
Tdef= P + λ(I − P)
is an isometry for some λ ∈ T \ 1, then T is a surjectiveisometry and σ(T ) = 1, λ.
Conversely, if T is a surjective isometry with σ(T ) = 1, λ,λ 6= 1, then |λ| = 1 and
Pdef=
T − λI
1− λ
is a projection.
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Generalized bicircular projectionsand the spectrum of the corresponding isometry
If P is a projection such that
Tdef= P + λ(I − P)
is an isometry for some λ ∈ T \ 1, then T is a surjectiveisometry and σ(T ) = 1, λ.
Conversely, if T is a surjective isometry with σ(T ) = 1, λ,λ 6= 1, then |λ| = 1 and
Pdef=
T − λI
1− λ
is a projection.
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The spectrum of surjective isometries
Every isolated point in the spectrum σ(T ) of a surjective isometryT on a Banach space is an eigenvalue of T with a complementedeigenspace. In particular, if σ(T ) = λ0, λ1, . . . , λn−1 then all λi ’sare eigenvalues, and the associated eigenprojections Pi ’s satisfy
P0⊕P1⊕· · ·⊕Pn−1 = I and T = P0 +λ1P1 + · · ·+λn−1Pn−1.
Here, we write P ⊕ Q to indicate that the Banach spaceprojections P and Q disjoint from each other, i.e., PQ = QP = 0.
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Generalized n-circular projections
Definition
Let P0 be a nonzero projection on a Banach space X , and n ≥ 2.We call P0 a generalized n-circular projection if there exists a(surjective) isometry T : X → X with σ(T ) = 1, λ1, . . . , λn−1consisting of n distinct (modulus one) eigenvalues such that P0 isthe eigenprojection of T associated to λ0 = 1.In this case, there are nonzero projections P1, . . . ,Pn−1 on X suchthat
P0⊕P1⊕· · ·⊕Pn−1 = I and T = P0 +λ1P1 + · · ·+λn−1Pn−1.
We also say that P0 is a generalized n-circular projectionassociated with (λ1, . . . , λn−1,P1, . . . ,Pn−1).
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Generalized n-circular projections on C0(Ω)
Let Ω be a locally compact Hausdorff space.Let ϕ : Ω→ Ω be a homeomorphism with period m, i.e., ϕm = idΩ
and ϕk 6= idΩ for k = 1, 2, . . . ,m − 1.Let u be a continuous unimodular scalar function on Ω such that
u(ω) · · · u(ϕm−1(ω)) = 1, ω ∈ Ω.
Then the surjective isometry T : C0(Ω)→ C0(Ω) defined by
Tf (ω) = u(ω)f (ϕ(ω))
satisfies T m = I .Therefore, the spectrum σ(T ) = λ0, λ1, . . . , λn−1 consists of ndistinct mth roots of unity.Replacing T with λ0T , we can assume that λ0 = 1.
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This gives rise to a spectral decomposition
I = P0 ⊕ P1 ⊕ · · · ⊕ Pn−1, T = λ0P0 + λ1P1 + · · ·+ λn−1Pn−1.
Here, the spectral projections are defined by
Pi f (w) =(I + λiT + · · ·+ λi
m−1T m−1)f (ω)
m
=1
m
(f (ω) + λiu(ω)f
(ϕ(ω)
)+ . . .
+ λim−1
u(ω) . . . u(ϕm−2(ω)
)f(ϕm−1(ω)
))for all f ∈ C0(Ω), ω ∈ Ω, and i = 0, 1, . . . , n − 1.An mth root λ of unity does not belong to σ(T ) if and only if
I + λT + · · ·+ λm−1
T m−1 = 0.
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Theorem (D. I. , C.-N. Liu and N.-C. Wong)
Let Ω be a connected locally compact space. Let T be a surjectiveisometry of C0(Ω) with finite spectrum consisting of n points.Then all eigenvalues of T are of finite orders.
Definition
We call the generalized n-circular projection P0 periodic (resp.primitive) if it is an eigenprojection of a periodic surjectiveisometry T of period m ≥ n (resp. of period m = n).
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Primitive generalized n-circular projections on C (T)
Example
Let n be a positive integer and let τ = e i 2πn .
Let T be the unit circle in the complex plane.Then Tf (z) = f (τz) is a surjective isometry of C (T), and
σ(T ) = 1, τ, . . . , τn−1.
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Generalized bicircular and tricircular projections on C0(Ω)
Theorem
Let Ω be a connected compact Hausdorff space and letP0 : C0(Ω)→ C0(Ω) be a projection. Then the following holds.
(i) [F. Botelho, 2008]If T = P0 + λ1P1, with P0 ⊕ P1 = I , is an isometry for someλ1 ∈ T \ 1 then σ(T ) = 1,−1.
(ii) [A. B. Abubaker and S. Dutta, 2011]If T = P0 + λ1P1 + λ2P2, with P0 ⊕ P1 ⊕ P2 = I , is anisometry for some distinct λ1, λ2 ∈ T \ 1 then
σ(T ) = 1, e i 2π3 , e i 4π
3 .
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Generalized 4-circular projections on C0(Ω) – an example
Example
A = (x , y , z) ∈ R3 : x , y , z ∈ [0, 1],B = (s,−s, 0) ∈ R3 : s ∈ [−1, 1], Ω = A ∪ B.
ϕ(x , y , z) =
(y , z , x), if (x , y , z) ∈ A;(−x ,−y ,−z), if (x , y , z) ∈ B.
The isometry Tfdef= f ϕ of period 6 has 4 eigenvalues
λ0 = 1, λ1 = −1, λ2 = β, λ3 = β2, where β = e i 2π3 .
Hence T = P0 − P1 + βP2 + β2P3.
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Generalized n-circular projections on C0(Ω) – the structure theorem
Theorem (D. I. , C.-N. Liu and N.-C. Wong)
Let Ω be a connected locally compact Hausdorff space.Let ϕ : Ω→ Ω be a homeomorphism and u be a unimodularcontinuous scalar function defined on Ω.Let P0 be a generalized n-circular projection on C0(Ω) associatedto Tf = u · f ϕ with the spectral decomposition
I = P0 ⊕ P1 ⊕ · · · ⊕ Pn−1,
T = P0 + λ1P1 + · · ·+ λn−1Pn−1.
Assume all eigenvalues λ0 = 1, λ1, . . . , λn−1 of T have a(minimum) finite common period m ≥ n.In particular, all of them are mth roots of unity, and T m = I .Then the following holds.
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Theorem (continuation)
The homeomorphism ϕ has (minimum) period m.
The cardinality k(ω) of the orbit ω, ϕ(ω), ϕ2(ω), . . . of eachpoint ω under ϕ is not greater than n.
m is the least common multiple of k(ω) for all ω in Ω.
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Theorem (continuation)
The spectrum σ(T ) of T can be written as a union of thecomplete set of k(ω)th roots of the modulus one scalarαω = u(ω)u(ϕ(ω)) · · · u(ϕk(ω)−1(ω)). More precisely,
σ(T ) =⋃ω∈Ω
λω, λωηω, λωη2ω, . . . , λωη
k(ω)−1ω ,
where λω and ηω are primitive k(ω)th roots of αω and unity,respectively. We call the set in the union a complete cycle ofk(ω)th roots of unity shifted by λω.
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Theorem (continuation)
If u(ω) = 1 on Ω then we can choose all λω = 1, and thusσ(T ) consists of all k(ω)th roots of unity.
If m is a prime integer, then n = m and σ(T ) consists of thecomplete cycle of nth roots of unity.
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Corollary
Let Ω be a connected locally compact Hausdorff space. Then everygeneralized bicircular or tricircular projection P0 on C0(Ω) isprimitive. In other words, P0 can only be an eigenprojection of asurjective isometry T on C0(Ω) with a spectral decomposition
T = P0 − (I − P0) for the bicircular case,
T = P0 + βP1 + β2P2 for the tricircular case,
where β = e i 2π3 .
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Corollary
Let Ω be a connected locally compact Hausdorff space.Let Tf = u · f ϕ be a surjective isometry on C0(Ω) with thespectral decomposition
T = P0 + λ1P1 + λ2P2 + λ3P3.
Then σ(T ) = 1, λ1, λ2, λ3 can only be one of the following:
1,−1, i ,−i, 1,−1, β, β2, 1,−1,−β,−β2,
1,−β, β, β2, 1, β, β2,−β2.
All above cases can happen. Here β = e i 2π3 .
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Corollary
Let Ω be a connected locally compact Hausdorff space.Let Tf = u · f ϕ be a surjective isometry on C0(Ω) with thespectral decomposition
T = P0 + λ1P1 + λ2P2 + λ3P3 + λ4P4.
Then σ(T ) = 1, λ1, λ2, λ3, λ4 can only be one of the following:
1, δ, δ2, δ3, δ4, 1,−1, β,−β, β2, 1,−1, β,−β,−β2,
1,−1, β, β2,−β2, 1, β,−β, β2,−β2.
All above cases can happen. Here, β = e i 2π3 and δ = e i 2π
5 .If u is a constant function, then only the first case is allowed.
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Generalized 5-circular projections on C0(Ω) – an example
Example
A1 = (1, 0, ρ) ∈ R3 : ρ ∈ [0, π],
A2 = (1,2π
3, ρ) ∈ R3 : ρ ∈ [0, π],
A3 = (1,4π
3, ρ) ∈ R3 : ρ ∈ [0, π],
B = (r , 0, 0) ∈ R3 : r ∈ [1/2, 3/2],C = (r , 0, π) ∈ R3 : r ∈ [1/2, 3/2],Ω = A1 ∪ A2 ∪ A3 ∪ B ∪ C .
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Generalized 5-circular projections on C0(Ω) – an example
Example
ϕ(r , θ, ρ) =
(r , θ + 2π
3 , ρ), if (r , θ, ρ) ∈ A1 ∪ A2 ∪ A3;(2− r , θ, ρ), if (r , θ, ρ) ∈ B ∪ C .
u(r , θ, ρ) =
e i 2ρ
3 , if (r , θ, ρ) ∈ A1 ∪ A2;
e−i 4ρ3 , if (r , θ, ρ) ∈ A3;
1, if (r , θ, ρ) ∈ B;
e i 2π3 , if (r , θ, ρ) ∈ C .
Then Tfdef= u · f ϕ has period 6.
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Example
σ(T ) = 1, β, β2 ∪ 1,−1 ∪ β,−β = 1,−1, β,−β, β2.
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Non-primitive generalized n-circular projections on C0(Ω)
Theorem (D. I. , C.-N. Liu and N.-C. Wong)
There exists a non-primitive generalized n-circular projection oncontinuous functions on a connected compact Hausdorff space foreach n ≥ 4.
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Generalized n-circular projections on JB*-triples
Theorem (D. I., 2017)
Let A be a JB*-triple, and P0 : A → A be a generalized n-circularprojection, n ≥ 2, associated with (λ1, . . . , λn−1,P1, . . . ,Pn−1).Let λ0 = 1. Then one of the following holds.
(i) There exist i , j , k ∈ 0, 1, . . . , n − 1, k 6= i , k 6= j , such thatλiλjλk ∈ λm : m = 0, 1, . . . , n − 1.
(ii) All P0, P1, . . . , Pn−1 are hermitian.
When n = 2: if P is not hermitian then λ2 ∈ 1, λ, or λ ∈ 1, λ;hence λ = −1.
When n = 3: if P,Q,R are not hermitian then λ1λ2 = 1, orλ2
1 = λ2, or λ22 = λ1.
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Theorem (D. I., 2017)
Let A be a JB*-triple, and P0 : A → A be a generalized n-circularprojection, n ≥ 2, associated with (λ1, . . . , λn−1,P1, . . . ,Pn−1).Let λ0 = 1. Then one of the following holds.
(i) There exist i , j , k ∈ 0, 1, . . . , n − 1, k 6= i , k 6= j , such thatλiλjλk ∈ λm : m = 0, 1, . . . , n − 1.
(ii) All P0, P1, . . . , Pn−1 are hermitian.
When n = 2: if P is not hermitian then λ2 ∈ 1, λ, or λ ∈ 1, λ;hence λ = −1.
When n = 3: if P,Q,R are not hermitian then λ1λ2 = 1, orλ2
1 = λ2, or λ22 = λ1.
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Generalized n-circular projections on JB*-triples
Theorem (D. I., 2017)
Let A be a JB*-triple, and P0 : A → A be a generalized n-circularprojection, n ≥ 2, associated with (λ1, . . . , λn−1,P1, . . . ,Pn−1).Let λ0 = 1. Then one of the following holds.
(i) There exist i , j , k ∈ 0, 1, . . . , n − 1, k 6= i , k 6= j , such thatλiλjλk ∈ λm : m = 0, 1, . . . , n − 1.
(ii) All P0, P1, . . . , Pn−1 are hermitian.
When n = 2: if P is not hermitian then λ2 ∈ 1, λ, or λ ∈ 1, λ;hence λ = −1.
When n = 3: if P,Q,R are not hermitian then λ1λ2 = 1, orλ2
1 = λ2, or λ22 = λ1.
Dijana Ilisevic On isometries on some Banach spaces