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1.
The characteristic function has been an important tool for studying completely non-unitary contractions on Hilbert spaces.
In this note, we consider completely non-coisometric contractive tuples of commuting operators on a Hilbert space H. We show
that the characteristic function, which is now an operator-valued analytic function on the open Euclidean unit ball in ℂn, is a complete unitary invariant for such a tuple. We prove that the characteristic function satisfies a natural transformation
law under biholomorphic mappings of the unit ball. We also characterize all operator-valued analytic functions which arise
as characteristic functions of pure commuting contractive tuples. 相似文献
2.
Given Banach spaces X and Y, we show that, for each operator-valued analytic map ${\alpha \in \mathcal O (D,\mathcal L(Y,X))}$ satisfying the finiteness condition ${\dim (X/\alpha (z)Y) < \infty}$ pointwise on an open set D in ${\mathbb {C}^n}$ , the induced multiplication operator ${\mathcal O(U,Y) \stackrel{\alpha}{\longrightarrow} \mathcal O (U,X)}$ has closed range on each Stein open set ${U \subset D}$ . As an application we deduce that the generalized range ${{\rm R}^{\infty}(T) = \bigcap_{k \geq 1}\sum_{| \alpha | = k} T^{\alpha}X}$ of a commuting multioperator ${T \in \mathcal L(X)^n}$ with ${\dim(X/\sum_{i=1}^n T_iX) < \infty}$ can be represented as a suitable spectral subspace. 相似文献
3.
In 1959 E. Bishop used a Banach-space version of the analyticduality principle established by e Silva, Köthe, Grothendieckand others to study connections between spectral decompositionproperties of a Banach-space operator and its adjoint. Accordingto Bishop a continuous linear operator T L(X) on a Banach spaceX satisfies property (rß) if the multiplication operator is injective with closed range for each open set U in the complex plane. In the present articlethe analytic duality principle in its original locally convexform is used to develop a complete duality theory for property(rß). At the same time it is shown that, up to similarity,property (rß) characterizes those operators occurringas restrictions of operators decomposable in the sense of C.Foias, and that its dual property, formulated as a spectraldecomposition property for the spectral subspaces of the givenoperator, characterizes those operators occurring as quotientsof decomposable operators. It is proved that, unlike the situationfor commuting subnormal operators, each finite commuting systemof operators with property (rß) can be extended toa finite commuting system of decomposable operators. Meanwhilethe results of this paper have been used to prove the existenceof invariant subspaces for subdecomposable operators with sufficientlyrich spectrum. 1991 Mathematics Subject Classification: 47A11,47B40. 相似文献
4.
Let be a unital closed subalgebra of the algebra of all continuous functions on a compact set in . We define the notion of an -isometry and show that, under a suitable regularity condition needed to apply Aleksandrov's work on the inner function problem, every -isometry is reflexive. This result applies to commuting isometries, spherical isometries, and more generally, to all subnormal tuples with normal spectrum contained in the Bergman-Shilov boundary of a strictly pseudoconvex or bounded symmetric domain.
5.
LetT L(X) be a continuous linear operator on a complex Banach spaceX. We show thatT possesses non-trivial closed invariant subspaces if its localizable spectrum loc(T) is thick in the sense of the Scott Brown theory. Since for quotients of decomposable operators the spectrum and the localizable spectrum coincide, it follows that each quasiaffine transformation of a Banach-space operator with Bishop's property () and thick spectrum has a non-trivial invariant subspace. In particular it follows that invariant-subspace results previously known for restrictions and quotients of decomposable operators are preserved under quasisimilarity. 相似文献
6.
A theorem of Sz.-Nagy and Foias [9] shows that the characteristic function
of a completely non-unitary contraction T is a complete unitary invariant for T. In this note we extend this theorem to the case of a pure commuting contractive tuple using a natural generalization of the characteristic function to an operator-valued analytic function defined on the open unit ball of
This function is related to the curvature invariant introduced by Arveson [3]. 相似文献
7.
Jörg Eschmeier 《Archiv der Mathematik》1989,52(6):562-570
8.
Jörg Eschmeier 《Integral Equations and Operator Theory》2007,59(2):165-172
The question whether every subnormal tuple
on a complex Hilbert space is reflexive is one of the major open problems in multivariable invariant subspace theory. Positive
answers have been given for subnormal tuples with rich spectrum in the unit polydisc or the unit ball. The ball case has been
extended by Didas [6] to strictly pseudoconvex domains. In the present note we extend the polydisc case by showing that every
subnormal tuple with pure components and rich Taylor spectrum in a bounded polydomain
is reflexive. 相似文献
9.
Jörg Eschmeier 《Integral Equations and Operator Theory》2011,69(2):171-182
Let
M ì H(\mathbbB){M \subset H(\mathbb{B})} be a homogeneous submodule of the n-shift Hilbert module on the unit ball in
\mathbbCn{\mathbb{C}^{n}}. We show that a modification of an operator inequality used by Guo and Wang in the case of principal submodules is equivalent
to the existence of factorizations of the form [Mzj*,PM] = (N+1)-1Aj{[M_{z_j}^*,P_M] = (N+1)^{-1}A_j}, where N is the number operator on
H(\mathbbB){H(\mathbb{B})}. Thus a proof of the inequality would yield positive answers to conjectures of Arveson and Douglas concerning the essential
normality of homogeneous submodules of
H(\mathbbB){H(\mathbb{B})}. We show that in all cases in which the conjectures have been established the inequality holds and leads to a unified proof
of stronger results. 相似文献
10.