Taylor Spectrum Research Articles

The Słodkowski spectra and higher Shilov boundaries

We investigate relations between the spectra defined by Słodkowski [14] and higher Shilov boundaries of the Taylor spectrum. The results generalize the well-known relation between the approximate point spectrum and the usual Shilov boundary.

On the joint spectrum and 𝐻^{∞}-functional calculus for pairs of commuting contractions

In this paper we show the existence of a pair of commuting completely nonunitary contractions ( S , T ) (S,T) on a Hilbert space, whose joint Taylor spectrum contains the torus, such that there is a bounded analytic function h h on the bidisk with h ( S , T ) = 0 h(S,T) = 0 .

Analytic Perturbation of the Taylor Spectrum

Let ${T_1}(z), \ldots ,{T_m}(z)$, $z \in G \subset {{\mathbf {C}}^k}$, be analytic families of bounded operators in a complex Banach space $X$, such that for each $z \in G$ the operators ${T_i}(z)$ and ${T_j}(z)$, $i,j = 1, \ldots ,n$, commute. Main result: If $K(z)$ denotes the Taylor spectrum of the tuple $({T_1}(z), \ldots ,{T_m}(z))$, then the set-valued function $K:G \to {2^{{\mathbf {C}}m}}$ is analytic. Analyticity of such set-valued functions is defined here by a simultaneous local maximum property of $k$-tuples of complex polynomials on the graph of $K$.

Analytic perturbation of the Taylor spectrum

Let T 1 ( z ) , … , T m ( z ) {T_1}(z), \ldots ,{T_m}(z) , z ∈ G ⊂ C k z \in G \subset {{\mathbf {C}}^k} , be analytic families of bounded operators in a complex Banach space X X , such that for each z ∈ G z \in G the operators T i ( z ) {T_i}(z) and T j ( z ) {T_j}(z) , i , j = 1 , … , n i,j = 1, \ldots ,n , commute. Main result: If K ( z ) K(z) denotes the Taylor spectrum of the tuple ( T 1 ( z ) , … , T m ( z ) ) ({T_1}(z), \ldots ,{T_m}(z)) , then the set-valued function K : G → 2 C m K:G \to {2^{{\mathbf {C}}m}} is analytic. Analyticity of such set-valued functions is defined here by a simultaneous local maximum property of k k -tuples of complex polynomials on the graph of K K .

On closures of joint similarity orbits

For an n-tuple T=(T1,..., Tn) of operators on a Hilbert spacexxHx, the joint similarity orbit of T isxxSx(T)={VTV−1 =(VT1V−1,...,VTnV−1): V is invertible onxxHx}. We study the structure of the norm closure ofxxSx, both in the case when T is commutative and when it is not. We first develop a Rota-model for the Taylor spectrum and use it to study n-tuples with totally disconnected Taylor spectrum, in particular quasinilpotent ones. We then consider limits of nilpotent n-tuples, and of normal n-tuples. For noncommuting n-tuples, we present a number of surprising facts relating the closure ofxxSx(T) to the Harte spectrum of T and the lack of commutativity of T. We show that a continuous function which is constant onxxSx(T) for all T must be constant. We conclude the paper with a detailed study of closed similarity orbits.

The Taylor spectrum and quotient Banach spaces

The Taylor spectrum and quotient Banach spaces

Spectral inclusion for doubly commuting subnormal 𝑛-tuples

Let S = ( S 1 , … , S n ) S = ({S_1}, \ldots ,{S_n}) be a doubly commuting n n -tuple of subnormal operators on a Hilbert space H \mathcal {H} and N = ( N 1 , … , N n ) N = ({N_1}, \ldots ,{N_n}) be its minimal normal extension acting on a Hilbert space K ⊃ H \mathcal {K} \supset \mathcal {H} . We show that Sp ( S , H ) ⊃ Sp ( N , K ) \text {Sp}(S,\mathcal {H}) \supset \text {Sp}(N,\mathcal {K}) and Sp ( S , H ) ⊂ p .c .h .(Sp( N , K ) ) {\text {Sp}(S,\mathcal {H}) \subset \text {p}\text {.c}{\text {.h}}\text {.(Sp(}}N,\mathcal {K})) , where Sp {\text {Sp}} denotes Taylor spectrum and p.c.h. polynomially convex hull.

On the Localization of the Spectrum for Systems of Operators

Let a = (a,.., a) be a commuting system of linear continuous operators on a complex Banach space X. We show that, for any x E X, the local analytic spectrum a(a, x) [1] is contained in the spectral hull of the local spectrum sp(a, x) [4]. Introduction. In this paper we shall define more accurately the relation between two notions of local spectrum for systems of operators. Let X be a complex Banach space and let a = (a, . . . , an) be a commuting system (n-tuple) of linear continuous operators on X. The simplest way to define the spectrum of an arbitrary element x E X, with respect to a, is the following. Consider the union p(a, x) of all open sets D C C' such that there exist n analytic functions fl, ... ,fn: Df, satisfying X7.= I(zi ai)f (z) x, z E D. Then the spectrum of x with respect to a is the set a(a, x) = C' p(a, x). This notion of local spectrum was used in [1] for the study of spectral decompositions dependent on functional calculi. In [4] the author has studied spectral decompositions not necessarily dependent on functional calculi and has found very useful to define another notion of local spectrum. Generally speaking, the spectrum of x with respect to a in our sense, denoted by sp(a, x), is the support of a certain differential form (or rather of a cohomology class of such forms). This notion was suggested by a new definition of the functional analytic calculus for several commuting operators, proposed in [7]. We have proved in [4] that sp(a, x) c a(a, x) for any x E X. The main result of the present paper is that a(a, x) is contained in the spectral hull of sp(a, x), for any x E X. 1. Preliminaries. We shall recall the definition of the Cauchy-Weil integral which will be our main tool in what follows (for details, see [7, ?3] and [4, Preliminaries]). Let X be a complex Banach space and a = (a,, .. an) be a commuting system of linear continuous operators on X. Denote by sp(a, X) the Taylor spectrum of a on X [6]. Received by the editors May 17, 1977. AMS (MOS) subject classifications (1970). Primary 47D99; Secondary 32C30, 32C35.

The Taylor spectrum and spectral decompositions

The Taylor spectrum and spectral decompositions