Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections On isometries on some Banach spaces – something old, something new, something borrowed, something blue, Part I Dijana Iliˇ sevi´ c University of Zagreb, Croatia Recent Trends in Operator Theory and Applications Memphis, 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 Iliˇ sevi´ c On isometries on some Banach spaces
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On isometries on some Banach spacesfunctions on compact metric spaces. Stone (1937): for real-valued functions on compact Hausdor spaces. Dijana Ili sevi c On isometries on some Banach
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IntroductionIsometries on some important Banach spaces
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)∗.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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)∗.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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).
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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).
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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).
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Generalized n-circular projections on C0(Ω) – the structure theorem
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.
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
Generalized bicircular and tricircular projections on C0(Ω)
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 .
Dijana Ilisevic On isometries on some Banach spaces
IntroductionIsometries on some important Banach spaces
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
IntroductionIsometries on some important Banach spaces
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
IntroductionIsometries on some important Banach spaces
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