On isometries on some Banach spacesfunctions on compact metric spaces. Stone (1937): for real-valued...

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



Something old, something new,

something borrowed, something blue

is referred to the collection of items that

helps to guarantee fertility and prosperity.




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.




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.




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.




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).




Trivial isometries







Trivial isometries







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 .




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.




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.




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

).




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 .




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.




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.




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.




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.




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.




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.




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 .




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.




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.




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.




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 .




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 .




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).




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).




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).




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.




Hermitian projections

By a projection on a complex Banach space we mean a linearoperator P such that P2 = P.


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.




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.




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.




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.


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).




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.


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).




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).




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).




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.


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).




Generalized bicircular projections on Kn(C)


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.




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)∗.




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.




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 .




Applications to some important JB*-triples

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)∗.




Generalized bicircular projections on C0(Ω)

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.




Generalized bicircular projections on C0(Ω)


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.




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).




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.




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.




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.




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).




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.





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.





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).




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.




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 .




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.




Generalized n-circular projections on C0(Ω) – the structure theorem


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.





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.






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.






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.




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 .




Generalized 4-circular projections on C0(Ω)

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 .




Generalized 5-circular projections on C0(Ω)

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.





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 .





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.





Example

σ(T ) = 1, β, β2 ∪ 1,−1 ∪ β,−β = 1,−1, β,−β, β2.




Non-primitive generalized n-circular projections on C0(Ω)


There exists a non-primitive generalized n-circular projection oncontinuous functions on a connected compact Hausdorff space foreach n ≥ 4.




Generalized n-circular projections on JB*-triples


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.











1 = λ2, or λ22 = λ1.











1 = λ2, or λ22 = λ1.


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