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1.
Aluthge transforms of operators 总被引:7,自引:0,他引:7
Associated with every operatorT on Hilbert space is its Aluthge transform
(defined below). In this note we study various connections betweenT and
, including relations between various spectra, numerical ranges, and lattices of invariant subspaces. In particular, we show that if
has a nontrivial invariant subspace, then so doesT, and we give various applications of our results. 相似文献
2.
Fernanda Botelho 《Linear algebra and its applications》2010,432(1):275-282
We characterize essential normality for certain elementary operators acting on the Hilbert-Schmidt class. We find the Aluthge transform of an elementary operator of length one. We show that the Aluthge transform of an elementary 2-isometry need not be a 2-isometry. We also characterize hermitian elementary operators of length 2. 相似文献
3.
On the Range of the Aluthge Transform 总被引:1,自引:0,他引:1
Let
be the algebra of all bounded linear operators on a complex separable Hilbert space
For an operator
let
be the Aluthge transform of T and we define
for all
where T = U|T| is a polar decomposition of T. In this short note, we consider an elementary property of the range
of Δ. We prove that R(Δ) is neither closed nor dense in
However R(Δ) is strongly dense if
is infinite dimensional.
An erratum to this article is available at . 相似文献
4.
In this note we give an example of an ∞-hyponormal operator T whose Aluthge transform
is not (1+ɛ)-hyponormal for any ɛ > 0 and show that the sequence
of interated Aluthge transforms of T need not converge in the weak operator topology, which solve two problems in [6]. 相似文献
5.
We prove that every one dimensional extension of a separably acting normal operator has a cyclic commutant, and that every non-algebraic normal operator has a two-dimensional extension which fails to have a cyclic commutant. Contrasting this, we prove that ifT is an extension of a normal operator by an algebraic operator then the weakly closed algebraW(T) has a separating vector.Partially supported by NSF Grant DMS-9107137 相似文献
6.
A continuation of the study of thew-hyponormal operators is presented. It is shown thatw-hyponormal operators are paranormal. Sufficient conditions which implyw-hyponormal operators are normal are given. The nonzero points of the approximate and joint approximate point spectra are shown to be identical forw-hyponormal operators. The square of an invertiblew-hyponormal operator is shown to bew-hyponormal. 相似文献
7.
LetH be a complex infinite-dimensional separable Hilbert space. An operatorT inL(H) is called totally P-posinormal (see [9]) iff there is a polynomialP with zero constant term such that
for each
, whereT
z
=T–zI andM(z) is bounded on the compacts of C. In this paper we prove that every totally P-posinormal operator is subscalar, i.e. it is the restriction of a generalized scalar operator to an invariant subspace. Further, a list of some important corollaries about Bishop's property and the existence of invariant subspaces is presented. 相似文献
8.
Eungil Ko 《Integral Equations and Operator Theory》2007,57(4):567-582
The backward Aluthge iterate (defined below) of a hyponormal operator was initiated in [11]. In this paper we characterize
the backward Aluthge iterate of a weighted shift. Also we show that the backward Aluthge iterate of a hyponormal operator
has an analogue of the single valued extension property for
. Finally, we show that backward Aluthge iterates of a hyponormal operator have scalar extensions. As a corollary, we get
that the backward Aluthge iterate of a hyponormal operator has a nontrivial invariant subspace if its spectrum has interior
in the plane. 相似文献
9.
Analysis of Non-normal Operators via Aluthge Transformation 总被引:1,自引:0,他引:1
Let T be a bounded linear operator on a complex Hilbert space
. In this paper, we show that T has Bishops property () if and only if its Aluthge transformation
has property (). As applications, we can obtain the following results. Every w-hyponormal operator has property (). Quasi-similar w-hyponormal operators have equal spectra and equal essential spectra. Moreover, in the last section, we consider Chs problem that whether the semi-hyponormality of T implies the spectral mapping theorem Re(T) = (Re T) or not. 相似文献
10.
《Mathematische Nachrichten》2017,290(17-18):2925-2933
In this paper we consider the following question. When does there exist a square root of a probability measure supported on ? This question is naturally related to subnormality of weighted shifts. The main result of this paper is that if μ is a finitely atomic probability measure having at most 4 atoms, then μ has a square root, i.e., there exists a measure ν such that (* means the convolution) if and only if the Aluthge transform of a subnormal weighted shift with Berger measure μ is subnormal. As an application of them, we give non‐trivial, large classes of probability measures having a square root. We also prove that there are 6 and 7‐atomic probability measures which don't have any square root. Our results have a connection to the following long‐open problem in Operator Theory: characterize the subnormal operators having a square root. 相似文献
11.
We investigate the spectral theory of the following general nonautonomous evolution equation
12.
Shanli Sun 《Linear algebra and its applications》2010,433(4):737-749
In this paper, the concept of generalized hermitian operators defined on a complex Hilbert space is introduced. It is shown that the spectrums and the Fredholm fields of generalized hermitian operators are both symmetric with respect to the real axis. Some other results on generalized hermitian operators are obtained. 相似文献
13.
Kehe Zhu 《Integral Equations and Operator Theory》2001,40(2):244-255
Supposef is a bounded analytic function in the unit disk andM
f
is the multiplication operator on an invariant subspaceI of the Bergman space. We show that wheren=dim(IzI). 相似文献
14.
Nathan S. Feldman 《Integral Equations and Operator Theory》2000,37(4):402-422
We study pure subnormal operators whose self-commutators have zero as an eigenvalue. We show that various questions in this are closely related to questions involving approximation by functions satisfying
and to the study ofgeneralized quadrature domains.First some general results are given that apply to all subnormal operators within this class; then we consider characterizing the analytic Toeplitz operators, the Hardy operators and cyclic subnormal operators whose self-commutators have zero as an eigenvalue. 相似文献
15.
We prove a stability theorem for the eigenvalues of general non-negative self-adjoint linear operators with compact resolvents and by applying it we prove a sharp stability result for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Neumann boundary conditions upon domain perturbation. 相似文献
16.
In this paper, we will give a simple proof of Theorem of [8] and some applications.This research is partially supported by Grant-in-Aid for Scientific Research. 相似文献
17.
B. P. Duggal 《Integral Equations and Operator Theory》1996,26(3):338-345
IfA
i
i=1, 2 are quasi-similarp-hyponormal operators such thatUi is unitary in the polar decompositionA
i
=U
i
|A
i
|, then (A
1)=(A
2) and
c(A1) = e(A2). Also a Putnam-Fuglede type commutativity theorem holds for p-hyponomral operators. 相似文献
18.
Jan Stochel 《Integral Equations and Operator Theory》2002,43(2):189-214
Various theorems on lifting strong commutants of unbounded subnormal (as well as formally subnormal) operators are proved. It is shown that the strong symmetric commutant of a closed symmetric operatorS lifts to the strong commutant of some tight selfadjoint extension ofS. Strong symmetric commutants of orthogonal sums of subnormal operators are investigated. Examples of (unbounded) irreducible subnormals, pure subnormals with rich strong symmetric commutants and cyclic subnormals with highly nontrivial strong commutants are discussed.This work was supported by the KBN grant # 2P03A 041 10. 相似文献
19.
Victor I. Burenkov 《Journal of Differential Equations》2008,244(7):1712-1740
We prove sharp stability results for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Dirichlet boundary conditions upon domain perturbation. 相似文献
20.
In this paper we discuss necessary conditions and sufficient conditions for the compression of an analytic Toeplitz operator onto a shift coinvariant subspace to have nontrivial reducing subspaces. We give necessary and sufficient conditions for the kernel of a Toeplitz operator whose symbol is the quotient of two inner functions to be nontrivial and obtain examples of reducing subspaces from these kernels. Motivated by this result we give necessary conditions and sufficient conditions for the kernel of a Toeplitz operator whose symbol is the quotient of two inner functions to be nontrivial in terms of the supports of the two inner functions. By studying the commutant of a compression, we are able to give a necessary condition for the existence of reducing subspaces on certain shift coinvariant subspaces. 相似文献