Operators shrinking the Arveson spectrum

Operators shrinking the Arveson spectrum A. R. Villena joint work with J. Alaminos and J. Extremera Departamento de Análisis Matemático Universidad de Granada BANACH ALGEBRAS 2009 STEFAN BANACH INTERNATIONAL MATHEMATICAL CENTER JULY 14-24, 2009 BEDLEWO, POLAND A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 1 / 31

The Arveson spectrum Let G be a locally compact abelian group, X be a complex Banach space, and τ : G inv ( B(X) ) be a bounded and strongly continuous group homomorphism. For every x X, the Arveson spectrum of x is defined as { } sp(x) = γ Ĝ : f (γ) = 0 for each f L 1 (G) with f (t)τ(t)xdt = 0. G A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 2 / 31

Examples Example (Support of the Fourier transform) Let G be a locally compact abelian group and let τ be the so-called regular representation of G on L 1 (G), τ : G B ( L 1 (G) ), [τ(t)f ](s) = f (t 1 s) (s, t G, f L 1 (G)). Then sp(f ) = supp f (f L 1 (G)). A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 3 / 31

Example (Local spectrum of an operator) Let X be a complex Banach space. Given an operator T B(X), the local resolvent set ρ(t, x) of T at the point x X is defined as the union of all open subsets U of C for which there is an analytic function f : U X which satisfies (T z)f (z) = x for each z U. The local spectrum σ(t, x) of T at x is then defined as σ(t, x) = C \ ρ(t, x). 1 Let T B(X) be a doubly power bounded operator and consider Then sp(x) = σ(t, x) τ : Z B(X), τ(k) = T k (k Z). (x X). 2 Let T B(X) be a hermitian operator and consider the one-parameter group Then sp(x) = σ(t, x) τ : R B(X), τ(t) = exp(itt ) (t R). (x X). A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 4 / 31

Example (Support of a spectral measure) Let A: D H H be a selfadjoint operator on a Hilbert space H. Then there is a projection-valued measure E on R such that A = + t de(t). For every x H, the map E( )x 2 defines a measure E x on the Borel subsets of R. Consider the one-parameter group τ : R U(H), τ(t) = exp(ita) (t R). Then sp(x) = supp ( E x ) (x H). A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 5 / 31

The problem In many different contexts we find out operators Φ: X Y, where X and Y are complex Banach spaces, which shrink the Arveson espectrum in the sense that sp(φx) sp(x) (x X) for appropriate representations τ X and τ Y of a locally compact abelian group G on X and Y, respectively. The problem A typical example of operator Φ shrinking the spectrum is the one intertwining τ X and τ Y Φ τ X (t) = τ Y (t) Φ (t G). The standard problem consists in determining whether every operator shrinking the Arveson spectrum necessarily intertwines the representations. A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 6 / 31

Our approach to the problem Suppose that Φ: X Y shrinks the Arveson spectrum. Pick t G and x X. We define a continuous bilinear map ϕ: A(T) A(T) Y, ϕ(f, g) = ( ( f (j)ĝ(k)τy (t j ) Φ τ X (t )x) ) k (f, g A(T)). j,k Z We check that sp(ϕ(f, g)) { } γ sp(x): γ(t) supp(f ) supp(g). A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 7 / 31

Our approach to the problem: Disjointness vanishing bilinear maps on A(T) Accordingly, ϕ satisfies the property f, g A(T), supp(f ) supp(g) = }{{ ϕ(f, g) = 0. } disjointness vanishing The bilinear map ϕ induces a continuous linear operator Ψ: A(T 2 ) X, Ψ(f ) = ( ) f (j, k)ϕ(z j, z k ) f A(T 2 ) with the property that j,k Z f A(T 2 ), supp(f ) {(z, z): z T} = }{{} Ψ(f ) = 0. At this point the synthesis is coming into the scene! A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 8 / 31

The set is a set of synthesis for A(T 2 ). This means that every function f A(T 2 ) vanishing at can be approximated by a sequence (f n ) in A(T 2 ) with the property that f n vanishes on a neighborhood of. We then consider the function f A(T 2 ) defined by f (z, w) = z w (z, w T) and there exists a sequence (f n ) in A(T 2 ) with Accordingly, we have f = lim f n and supp(f n ) = (n N). 0 = lim Ψ(f n ) = Ψ(f ) = ϕ(z, 1) ϕ(1, z). }{{}}{{} ( ) ( ) τ Y (t) Φx Φ τ X (t)x A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 9 / 31

Our contribution Theorem Let G be a locally compact abelian group. Let X and Y be Banach spaces and let τ X : G B(X) and τ Y : G B(Y ) bounded and strongly continuous group homomorphisms. If Φ B(X, Y ) shrinks the Arveson spectrum, i.e. then Φ intertwines τ X and τ Y, i.e. sp(φx) sp(x) (x X), Φ τ X (t) = τ Y (t) Φ (t G). A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 10 / 31

The secret of our success Theorem Let ϕ: A(T) A(T) X be a continuous bilinear map into some Banach space X with the property that f, g A(T), supp(f ) supp(g) = ϕ(f, g) = 0, then ϕ(z, 1) = ϕ(1, z). A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 11 / 31

The Arveson spectrum for unbounded representations The theory of the Arveson spectrum still works for a non-quasianalytic representation τ : G B(X). This means that the weight function ω given by ω(t) = max{1, τ(t) } (t G) satisfies the Beurling-Domar condition + k= log ω(t k ) 1 + k 2 < (t G). For the definition of the Arveson spectrum of τ at a point x X we now replace the group algebra L 1 (G) by the weighted group algebra L 1 (G, ω). { } sp(x) = γ Ĝ : f (γ) = 0 for each f L 1 (G, ω) with f (t)τ(t)xdt = 0. G A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 12 / 31

Our theorem may fail in the case where the representation is unbounded. Example Consider the representations τ 1, τ 2 : Z B(C 2 ) given by Then and τ 1 (k) = ( ) k 1 1 and τ 0 1 2 (k) = I C2 (k Z). sp τ2 (x) sp τ1 (x) (x C 2 ) I C 2 τ 1 (k) τ 2 (k) I C 2 (k 0). In this case we have 1 + k 2 τ 1 (k) 2 + k 2 (k Z). A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 13 / 31

For dealing with representations with polynomial growth we are required to involve disjointness vanishing bilinear maps on the weighted Fourier algebra { A α (T) = ϕ: A α (T) A α (T) X f C(T): f = k Z } f (k) (1 + k ) α < for some α 0. A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 14 / 31

Theorem Let ϕ: A α (T) A α (T) X be a continuous bilinear map into some Banach space X with the property that f, g A α (T), supp(f ) supp(g) = ϕ(f, g) = 0. Then for each N > 2α. N i=0 ( ) N ( 1) i ϕ(z N i, z i ) = 0 i This time, ϕ gives rise to a continuous linear operator Ψ: A 2α (T 2 ) X. However, the diagonal may fail to be a set of synthesis for A 2α (T 2 ) because of the weight. We are thus required to take care of the function f at which we intend to apply Ψ. The so-called Beurling-Pollard type theorems assert, roughly speaking, that a function f admits synthesis if its growth is appropriate. We show that this is just the case for the function f (z, w) = (z w) N. A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 15 / 31

An improvement to our contribution Theorem Let G be a locally compact abelian group. Let X and Y be Banach spaces and let τ X : G B(X) and τ Y : G B(Y ) strongly continuous group homomorphisms with polynomial growth, i.e. τy (t k ), τx (t k ) = O( k α ) as k for some α 0. If Φ B(X, Y ) is such that sp(φx) sp(x) (x X), then N i=0 whenever N > 2α. ( ) N ( 1) i τ Y (t N i ) Φ τ X (t i ) = 0 i (t G), A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 16 / 31

Applications: Translation invariant operators Corollary Let G be a locally compact abelian group. Then T : L 1 (G) L 1 (G) is translation invariant if and only if ) ) supp ( Tf supp( f (f L 1 (G)). Proof. We consider the regular representation of G on L 1 (G). Then ) ) supp ( Tf supp( f, } {{ } } {{ } sp(tf ) sp(f ) and therefore T shrinks the Arveson spectrum. A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 17 / 31

Applications: Determining operators through the spectrum Corollary (Colojoară-Foiaş, 1968) Let X be a complex Banach space and S, T B(X) be invertible operators with polynomial growth. Suppose that Then σ(s, x) σ(t, x) x X. N i=0 ( ) N ( 1) i T N i S i = 0, i whenever N > 2α, where α is such that S k, T k = O( k α ) as k Consequently, 1 If S and T are doubly power bounded, then S = T. 2 If S and T commute, then (S T ) N = 0. A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 18 / 31

Proof. We consider the representations τ S, τ T : Z B(X) given by τ S (k) = S k and τ T (k) = T k (k Z). Then σ(s, x) σ(t, x), }{{}}{{} sp τs (x) sp τt (x) which shows that the identity operator I X : X X shrinks the spectrum. Consequently, N i=0 ( ) N ( 1) i τ T (N i)i X τ S (i) i }{{} = 0 (t G), T N i S i A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 19 / 31

Corollary (I. Gelfand, 1941) Let T be a bounded linear operator on a complex Banach space X such that σ(t ) = {1}. If sup T k <, k Z then Example T = I X. It should be pointed out that Gelfand theorem fails in the case where the operator is not doubly power bounded. As a matter of fact, the operator T B(C 2 ) corresponding to the matrix ( ) 1 1 0 1 is different from the identity operator and nevertheless σ(t ) = {1}. In this case ( ) T k 1 k and 1 + k 0 1 2 T k 2 + k 2 k Z. A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 20 / 31

Corollary (E. Hille, 1944) Let T be a bounded linear operator on a complex Banach space X such that σ(t ) = {1}. If T k = O( k n ) as k for some n 0, then for each integer N with N > n. (T I X ) N = 0. Example It should be pointed out that both Gelfand and Hille theorems fail for comparing the given operator with an operator different from the identity. Indeed, let S, T B(C 2 ) be the operators with matrices respectively. Then ( ) 1 1 0 1 and T S is far from being nilpotent. and ( ) 1 0, 1 1 σ(t ) = σ(s) = {1} A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 21 / 31

Theorem (Local version of Gelfand-Hille theorem) Let X be a complex Banach space and let T be an invertible operator on X such that T k = O( k α ) as k for some α 0. If x X is such that for some λ C, then σ(t, x) = {λ} (T λi X ) N x = 0 whenever N N is such that N > 2α. In particular, if T is doubly power bounded, then Tx = λx. A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 22 / 31

Proof. First of all, it should be pointed out that λ σ(t, x) σ(t ) T. We define ϕ: A α (T) A α (T) X by ϕ(f, g) = g(λ) (k)t k Z f k x. Then ϕ is a disjointness vanishing continuous bilinear map. Our seminal result gives 0 = N i=0 ( ) N ( 1) i ϕ(z N i, z i ) = (T λi X ) N x. i }{{} λ i T N i x A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 23 / 31

Theorem (An improvement of the Colojoară-Foiaş theorem) Let X be a complex Banach space, S, T 1,..., T n, B(X) invertible operators with polynomial growth with the property that σ(s, x) n σ(t j, x) j=1 x X and that T 1,..., T n are pairwise commuting Then there exists N N such that C(S, T 1 ) N C(S, T n ) N I X = 0. Here the intertwiner C(S, T ) of S, T B(X) is defined by C(S, T ): B(X) B(X), C(S, T )A = SA AT A B(X). A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 24 / 31

Applications: Local operators on function spaces and its non-commutative versions Theorem (A. C. Zaanen, 1975) Let Ω be a locally compact Hausdorff space and let T : C 0 (Ω) C 0 (Ω) be a bounded linear operator with the property that f C 0 (Ω), E Ω, f = 0 on E Tf = 0 on E. }{{} supp(tf ) supp(f ) Then there exists g C b (Ω) such that Tf = fg f C 0 (Ω). A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 25 / 31

Theorem (Our non-commutative version) Let A be a (unital) C -algebra and let T : A A be a bounded linear operator with the property that x A, p projection in A, px = 0 }{{ ptx = 0. } left locality Then there exists a A such that Tx = xa x A. A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 26 / 31

Proof. For every unitary u A, the map ϕ: A(T) A(T) A defined by ϕ(f, g) = f (j)ĝ(k)u j T (u k ) j,k Z is disjointness vanishing. This entails that T (u) = ut (1). Since A is generated by the unitaries, it follows that T (x) = x T (1) }{{} a x A. A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 27 / 31

If we remove the requirement of A being unital, then the theorem still works though the element a defining the operator T lies in M(A), the multiplier algebra of A (i.e., the largest C -algebra containing A as an essential ideal). Of course, if A is unital, then M(A) = A. If A = C 0 (Ω), then M(A) = C b (Ω). If A = K(H), then M(A) = L(H). A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 28 / 31

Theorem (A. C. Zaanen, 1975) Let (Ω, Σ, µ) be a σ-finite measure space, 1 p, and let T : L p (µ) L p (µ) be a bounded linear operator with the property that f L p (µ), E Σ, f = 0 a.e. on E }{{ Tf = 0 a.e. on E. } locality Then there exists g L (µ) such that Tf = fg f L p (µ). A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 29 / 31

Theorem (Our noncommutative version) Let M be a von Neumann algebra with a (normal semifinite faithful) trace τ, 1 p, and let T : L p (M, τ) L p (M, τ) be a bounded linear operator with the property that x L p (M, τ), p projection in M, px = 0 ptx = 0. Then there exists a M such that Tx = xa x L p (M, τ). Let M = L (µ). Then integration with respect to µ gives a n.s.f. trace and L p (M) = L p (µ) (1 p ). Let M = L(H) and τ the usual trace on L(H). Then L p (M) is the Schatten class S p (H) (1 p < ). A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 30 / 31

Applications: Determining observables in quantum mechanics through the support 1 Every physical observable in quantum mechanics is mathematically represented by a selfadjoint operator. 2 The outcome of a measurement of an observable is of statistical nature. The value obtained in a measurement of the observable A on the state ψ is a random variable Âψ whose probability distribution is given by the spectral measure E ψ of A at ψ, so that the probability to obtain a value in R as the result of this measure is given by Corollary P [ Âψ ] = E( )ψ 2. Let A and B be observables on a quantum physical system with the property that ] ] P [Âψ = 1 P [ Bψ = 1. Then A = B. A. R. Villena (Granada) Operators shrinking the Arveson spectrum Banach Algebras 2009 31 / 31