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 Iliˇsevi´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 Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized 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 Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections Isometries Isometries are maps between metric spaces which preserve distance between elements. Definition Let (X ; j · j) and (Y; k · k) be two normed spaces over the same field. A linear map ': X!Y is called a linear isometry if k'(x)k = jxj; x 2 X : We shall be interested in surjective linear isometries on Banach spaces. One of the main problems is to give explicit description of isometries on a particular space. Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections Isometries Isometries are maps between metric spaces which preserve distance between elements. Definition Let (X ; j · j) and (Y; k · k) be two normed spaces over the same field. A linear map ': X!Y is called a linear isometry if k'(x)k = jxj; x 2 X : We shall be interested in surjective linear isometries on Banach spaces. One of the main problems is to give explicit description of isometries on a particular space. Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized 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 Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections Trivial isometries Trivial isometries are isometries of the form λI for some λ 2 T, where T = fλ 2 F : jλj = 1g. The spectrum of a surjective linear isometry is contained in T. For any Banach space X (real or complex) there is a norm k · k on X , equivalent to the original one, such that (X ; k · k) has only trivial isometries (K. Jarosz, 1988). Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections Trivial isometries Trivial isometries are isometries of the form λI for some λ 2 T, where T = fλ 2 F : jλj = 1g. The spectrum of a surjective linear isometry is contained in T. For any Banach space X (real or complex) there is a norm k · k on X , equivalent to the original one, such that (X ; k · k) has only trivial isometries (K. Jarosz, 1988). Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections Trivial isometries Trivial isometries are isometries of the form λI for some λ 2 T, where T = fλ 2 F : jλj = 1g. The spectrum of a surjective linear isometry is contained in T. For any Banach space X (real or complex) there is a norm k · k on X , equivalent to the original one, such that (X ; k · k) has only trivial isometries (K. Jarosz, 1988). Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections Norms induced by the inner product Let V be a finite dimensional vector space over F 2 fR; Cg, equipped with the norm k · k induced by the inner product hx; yi = tr (xy ∗) = y ∗x (Frobenius norm). Then U is a linear isometry on (V; k · k) if and only if the following holds. 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, Ut U = UUt = I : Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized 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 t V ; where U; V 2 Mn(C) are unitaries. Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized 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! UXV for some fixed unitary (orthogonal) U; V 2 Mn(F). A norm k · k on Mn(F) is called a unitarily invariant norm if kg(X )k = kX k for all g 2 G, X 2 Mn(F). If k · k is a unitarily invariant norm (which is not a multiple of the Frobenius norm) on Mn(F) 6= M4(R) then its isometry group is hG; τi, where τ : Mn(F) ! Mn(F) is the transposition operator. Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections Unitarily invariant norms on M4(F) In the case of M4(R) the isometry group is hG; τi or hG; τ; αi, with α: M4(R) ! M4(R) defined by α(X ) = (X + B1XC1 + B2XC2 + B3XC3)=2; where 1 0 0 −1 1 0 0 1 B = ⊗ ; C = ⊗ ; 1 0 1 1 0 1 0 −1 −1 0 0 −1 1 0 0 1 1 0 B = ⊗ ; C = ⊗ ; 2 1 0 0 −1 2 −1 0 0 1 0 −1 0 1 0 1 0 1 B = ⊗ ; C = ⊗ : 3 1 0 1 0 3 1 0 −1 0 Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized 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! Ut XU for some fixed unitary (orthogonal) U 2 Mn(F). A norm k · k on V 2 fSn(C); Kn(F)g is called a unitary congruence invariant norm if kg(X )k = kX k for all g 2 G, X 2 V . If k · k is a unitary congruence invariant norm on Sn(C), which is not a multiple of the Frobenius norm, then its isometry group is G. Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections Unitary congruence invariant norms on Kn(F) If k · k is a unitary congruence invariant norm on Kn(C), which is not a multiple of the Frobenius norm, then its isometry group is G if n 6= 4, and hG; γi if n = 4, where γ(X ) is obtained from X by interchanging its (1; 4) and (2; 3) entries, and interchanging its (4; 1) and (3; 2) entries accordingly. If k · k is a unitary congruence invariant norm on Kn(R), which is not a multiple of the Frobenius norm, then its isometry group is hG; τi if n 6= 4, and hG; τ; γi if n = 4. Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections Surjective linear isometries of C0(Ω) Let C0(Ω) be the algebra of all continuous complex-valued functions on a locally compact Hausdorff space Ω, vanishing at infinity. Theorem (Banach{Stone) Let T : C0(Ω1) ! C0(Ω2) be a surjective linear isometry. Then there exist a homeomorphism ':Ω2 ! Ω1 and a continuous unimodular function u :Ω2 ! C such that T (f )(!) = u(!)f '(!) ; f 2 C0(Ω1);! 2 Ω2: The first (Banach's) version of this theorem (1932): for real-valued functions on compact metric spaces. Stone (1937): for real-valued functions on compact Hausdorff spaces. Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections Surjective linear isometries of C0(Ω) Let C0(Ω) be the algebra of all continuous complex-valued functions on a locally compact Hausdorff space Ω, vanishing at infinity. Theorem (Banach{Stone) Let T : C0(Ω1) ! C0(Ω2) be a surjective linear isometry. Then there exist a homeomorphism ':Ω2 ! Ω1 and a continuous unimodular function u :Ω2 ! C such that T (f )(!) = u(!)f '(!) ; f 2 C0(Ω1);! 2 Ω2: The first (Banach's) version of this theorem (1932): for real-valued functions on compact metric spaces. Stone (1937): for real-valued functions on compact Hausdorff spaces. Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections C ∗-algebras A C ∗-algebra is a complex Banach ∗-algebra (A; k : k) such that ka∗ak = kak2 for all a 2 A: Example C = complex numbers, B(H) = all bounded linear operators on a complex Hilbert space H, K(H) = all compact operators on a complex Hilbert space H, C(Ω) = all continuous complex-valued functions on a compact Hausdorff space Ω, C0(Ω) = all continuous complex-valued functions on a locally compact Hausdorff space Ω, vanishing at infinity.