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1.
In the past several years, there has been considerable progress made on a general left-definite theory associated with a self-adjoint operator A that is bounded below in a Hilbert space H; the term ‘left-definite’ has its origins in differential equations but Littlejohn and Wellman [L. L. Littlejohn, R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181 (2) (2002) 280-339] generalized the main ideas to a general abstract setting. In particular, it is known that such an operator A generates a continuum {Hr}r>0 of Hilbert spaces and a continuum of {Ar}r>0 of self-adjoint operators. In this paper, we review the main theoretical results in [L. L. Littlejohn, R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181 (2) (2002) 280-339]; moreover, we apply these results to several specific examples, including the classical orthogonal polynomials of Laguerre, Hermite, and Jacobi. 相似文献
2.
Israel Feldman 《Integral Equations and Operator Theory》1993,16(3):385-391
Sufficient conditions are given for the finiteness of the discrete spectrum of the block Toeplitz operatorT
A generated in the spaceH
2
n
by self-adjoint matrix functionA(t)(|t|=1). These results are obtained by means of theorems concerning the spectrum of a perturbed self-adjoint operators. 相似文献
3.
Dominik Mielczarek 《Monatshefte für Mathematik》2008,154(2):157-171
In this paper we will prove that an averaging projection P
a
: K (H) → Y, given by the formula
, is the only norm-one projection. Here, K (H) is a space of compact operators on a separable real Hilbert space H, and Y is the subspace of K(H) consisting of all symmetric operators.
Author’s address: Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków,
Poland 相似文献
4.
The classical Adamjan-Arov-Krein (A-A-K) theorem relating the singular numbers of Hankel operators to best approximations of their symbols by rational functions is given an abstract version. This provides results for Hankel operators acting in weightedH
2(T; ), as well as inH
2(T
d
), and an A-A-K type extension of Sarason's interpolation theorem. In particular, it is shown that all compact Hankel operators inH
2(T
d
) are zero.Author partially supported by NSF grant DMS89-11717. 相似文献
5.
Differences of Composition Operators between Weighted Banach Spaces of Holomorphic Functions on the Unit Polydisk 总被引:1,自引:0,他引:1
Elke Wolf 《Results in Mathematics》2008,51(3-4):361-372
We consider differences of composition operators between given weighted Banach spaces H
∞
v
or H
0
v
of analytic functions defined on the unit polydisk D
N
with weighted sup-norms and give estimates for the distance of these differences to the space of compact operators. We also
study boundedness and compactness of the operators. This paper is an extension of [6] where the one-dimensional case is treated.
Received: May 15, 2007. Revised: October 8, 2007. 相似文献
6.
《Quaestiones Mathematicae》2013,36(1-3):229-256
Abstract This is a report on a number of recent results on composition operators which map, for 0 < p ? q ∞, the Hardy space Hp (on the unit disk in the complex plane) into H q. Attention is focused on questions of boundedness (existence), compactness, order boundedness and, in connection with the latter, on relating the absolutely summing and nuclearity character as well as special factorization properties of the operator to function theoretic properties of the defining symbol. Moreover, tools are provided to show that certain classes of operators can well be distinguished already on the level of composition operators. 相似文献
7.
Yang Changsen Li Haiying 《高校应用数学学报(英文版)》2006,21(1):64-68
The approximate point spectrum properties of p-ω-hyponormal operators are given and proved. In faet, it is a generalization of approximate point speetrum properties of ω- hyponormal operators. The relation of spectra and numerical range of p-ω-hyponormal operators is obtained, On the other hand, for p-ω-hyponormal operators T,it is showed that if Y is normal,then T is also normal. 相似文献
8.
Henrik Petersson 《Integral Equations and Operator Theory》2007,57(3):413-423
We prove that for any weighted backward shift B = Bw on an infinite dimensional separable Hilbert space H whose weight sequence w = (wn) satisfies
, the conjugate operator
is hypercyclic on the space S(H) of self-adjoint operators on H provided with the topology of uniform convergence on compact sets. That is, there exists an
such that
is dense in S(H). We generalize the result to more general conjugate maps
, and establish similar results for other operator classes in the algebra B(H) of bounded operators, such as the ideals K(H) and N(H) of compact and nuclear operators respectively. 相似文献
9.
The gap between hyponormal and subnormal Hilbert space operators can be studied using the intermediate classes of weakly n-hyponormal and (strongly) n-hyponormal operators. The main examples for these various classes, particularly to distinguish them, have been the weighted
shifts. In this paper we first obtain a characterization for a weakly n-hyponormal weighted shift Wα with weight sequence α, from which we extend some known results for quadratically hyponormal (i.e., weakly 2-hyponormal)
weighted shifts to weakly n-hyponormal weighted shifts. In addition, we discuss some new examples for weakly n-hyponormal weighted shifts; one illustrates the differences among the classes of 2-hyponormal, quadratically hyponormal,
and positively quadratically hyponormal operators. 相似文献
10.
Operator ranges and non-cyclic vectors for the backward shift 总被引:2,自引:0,他引:2
Michael Sand 《Integral Equations and Operator Theory》1995,22(2):212-231
In this paper we look at operators on the Hardy spaceH
2(D) with range containing all of the non-cyclic vectors of the backward shift. We show several classes of such operators must be surjective, including Toeplitz, Hankel and composition operators. 相似文献
11.
T. Len Miller Vivien G. Miller Michael M. Neumann 《Mediterranean Journal of Mathematics》2009,6(2):149-168
This paper centers on local spectral conditions that are both necessary and sufficient for the equality of the essential spectra
of two bounded linear operators on complex Banach spaces that are intertwined by a pair of bounded linear mappings. In particular,
if the operators T and S are intertwined by a pair of injective operators, then S is Fredholm provided that T is Fredholm and S has property (δ) in a neighborhood of 0. In this case, ind(T) ≤ ind(S), and equality holds precisely when the eigenvalues of the adjoint T* do not cluster at 0. By duality, we obtain refinements of results due to Putinar, Takahashi, and Yang concerning operators
with Bishop’s property (β) intertwined by pairs of operators with dense range. Moreover, we establish an extension of a result due to Eschmeier that,
under appropriate assumptions regarding the single-valued extension property, leads to necessary and sufficient conditions
for quasi-similar operators to have equal essential spectra. In particular it turns out that the single-valued extension property
plays an essential role in the preservation of the index in this context.
相似文献
12.
Pascal Lefèvre 《Integral Equations and Operator Theory》2009,63(4):557-569
We compute the essential norm of a composition operator relatively to the class of Dunford-Pettis operators or weakly compact
operators, on some uniform algebras of analytic functions. Even in the context of H∞ (resp. the disk algebra), this is new, as well for the polydisk algebras and the polyball algebras. This is a consequence
of a general study of weighted composition operators.
相似文献
13.
In this paper we relate the operators in the operator representations of a generalized Nevanlinna function N(z) and of the function −N(z)−1 under the assumption that z=∞ is the only (generalized) pole of nonpositive type. The results are applied to the Q-function for S and H and the Q-function for S and H∞, where H is a self-adjoint operator in a Pontryagin space with a cyclic element w, H∞ is the self-adjoint relation obtained from H and w via a rank one perturbation at infinite coupling, and S is the symmetric operator given by S=H∩H∞. 相似文献
14.
For the unilateral shift operator U on the Hardy space H2(T), we describe conditions on operators T, acting on H2(T), that are necessary and sufficient for the pair (U, T) to be jointly hyponormal. One necessary condition is that T be a Toeplitz operator. Consequently, we study certain nonanalytic symbols that give rise to Toeplitz operators hyponormal with the shift, and thereby obtain examples of noncommuting, jointly hyponormal pairs.Supported in part by a research grant from NSERC 相似文献
15.
The class of -hyponormal operators is introduced. This class contains allp-hyponormal operators. Certain properties of this class of operators are obtained. Among other things, it is shown that ifT is -hyponormal, then its spectral radius and norm are identical, and the nonzero points of its joint point spectrum and point spectrum are identical. Conditions under which a -hyponormal operator becomes normal, self-adjoint and unitary are given. 相似文献
16.
Yurij M. Berezansky Eugene W. Lytvynov Artem D. Pulemyotov 《Integral Equations and Operator Theory》2005,53(2):191-208
By definition, a Jacobi field
is a family of commuting selfadjoint three-diagonal operators in the Fock space
The operators J(ϕ) are indexed by the vectors of a real Hilbert space H+. The spectral measure ρ of the field J is defined on the space H− of functionals over H+. The image of the measure ρ under a mapping
is a probability measure ρK on T−. We obtain a family JK of operators whose spectral measure is equal to ρK. We also obtain the chaotic decomposition for the space L2(T−, dρ K). 相似文献
17.
Salah Mecheri 《Integral Equations and Operator Theory》2005,53(3):403-409
Let B(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. Let A = (A1,A2,.., An) and B = (B1, B2,.., Bn) be n-tuples in B(H), we define the elementary operator
by
In this paper we initiate the study of some properties of the range of such operators. 相似文献
18.
Fritz Gesztesy Mark Malamud Marius Mitrea Serguei Naboko 《Integral Equations and Operator Theory》2009,64(1):83-113
We study generalized polar decompositions of densely defined closed linear operators in Hilbert spaces and provide some applications
to relatively (form) bounded and relatively (form) compact perturbations of self-adjoint, normal, and m-sectorial operators.
Based upon work partially supported by the US National Science Foundation under Grant Nos. DMS-0400639 and FRG-0456306, and
the Austrian Science Fund (FWF) under Grant No. Y330. 相似文献
19.
《Quaestiones Mathematicae》2013,36(1-3):167-183
Abstract Since 1970 a number of operational quantities, characteristic of either the semi-Fredholm operators or of some “ideal” of compact-like operators, have been introduced in the theory of bounded operators between Banach spaces and applied successfully to for example perturbation theory. More recently such quantities have been introduced even in the abstract setting of Fredholm theory in a von Neumann algebra relative to some closed two-sided ideal. We show that in this fairly general setting there is only one “reasonable” set of such quantities—a result which in its present form is to the best of our knowledge new even in the case of B(H), the algebra of all bounded operators on a Hilbert space H. We accomplish this by first of all introducing the concept of a (reduced) minimum modulus in the setting of C*-algebras and developing the relevant techniques. In the process we generalise a result of Nikaido [N]. 相似文献
20.
R. C. Smith 《Integral Equations and Operator Theory》1996,25(3):329-335
Invertible composition operators on the Hardy spaceH
p
have automorphic symbols. For 1<p< andp2 it is shown that some elliptic composition operators are scalar while others are generalized scalar but not spectral, that parabolic composition operators are generalized scalar but not spectral and that hyperbolic composition operators do not have the single-valued extension property. 相似文献