Unitary equivalence and reductibility or invertibly weighted shifts

1971 ◽  
Vol 5 (2) ◽  
pp. 157-173 ◽  
Author(s):  
Alan Lambert
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Bounded Sequence ◽  
Complex Hilbert Space ◽  
Weighted Shifts ◽  
Infinite Dimensional ◽  
Reducing Subspaces ◽  
Positive Weights ◽  
Completely Reducible ◽  
Necessary And Sufficient

Let H be a complex Hilbert space and let {A1, A2, …} be a uniformly bounded sequence of invertible operators on H. The operator S on l2(H) = H ⊕ H ⊕ … given by S〈x0, x1, …〉 = 〈0, A1x0, A2x1, …〉 is called the invertibly veighted shift on l2(H) with weight sequence {An }. A matricial description of the commutant of S is established and it is shown that S is unitarily equivalent to an invertibly weighted shift with positive weights. After establishing criteria for the reducibility of S the following result is proved: Let {B1, B2, …} be any sequence of operators on an infinite dimensional Hilbert space K. Then there is an operator T on K such that the lattice of reducing subspaces of T is isomorphic to the corresponding lattice of the W* algebra generated by {B1, B2, …}. Necessary and sufficient conditions are given for S to be completely reducible to scalar weighted shifts.

scholarly journals Invariant subspace lattices of Lambert's weighted shifts

1982 ◽  
Vol 33 (1) ◽  
pp. 135-142
Author(s):  
B. S. Yadav ◽  
S. Chatterjee
Keyword(s):  
Hilbert Space ◽  
Invariant Subspace ◽  
Bounded Sequence ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Weighted Shifts ◽  
Positive Real ◽  
Subspace Lattice ◽  
Bounded Linear ◽  
Infinite Dimensional

AbstractLet B(H) be the Banach algebra of all (bounded linear) operators on an infinite-dimensional separable complex Hilbert space H and let be a bounded sequence of positive real numbers. For a given injective operator A in B(H) and a non-zero vector f in H, we put We define a weighted shift Tw with the weight sequence on the Hilbert space 12 of all square-summable complex sequences by . The main object of this paper is to characterize the invariant subspace lattice of Tw under various nice conditions on the operator A and the sequence .

scholarly journals Orbits tending to infinity under sequences of operators on Hilbert spaces

Filomat ◽  
2007 ◽  
Vol 21 (2) ◽  
pp. 161-171
Author(s):  
Sonja Mancevska ◽  
Marija Orovcanec
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Complex Hilbert Space ◽  
Infinite Dimensional ◽  
Set Of Points ◽  
Dense Set

In this paper are considered some sufficient conditions under which, for given sequence (Ti)i?1 of operators on an infinite-dimensional complex Hilbert space, there is a dense set of points whose orbits under each Ti tend strongly to infinity. .

scholarly journals On the equivalence of separability and extendability of quantum states

2017 ◽  
Vol 29 (04) ◽  
pp. 1750012 ◽  
Author(s):  
B. V. Rajarama Bhat ◽  
K. R. Parthasarathy ◽  
Ritabrata Sengupta
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Gaussian State ◽  
Bounded Operators ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Separability Property ◽  
Quantum De Finetti Theorem ◽  
Necessary And Sufficient ◽  
Symmetric States

Motivated by the notions of [Formula: see text]-extendability and complete extendability of the state of a finite level quantum system as described by Doherty et al. [Complete family of separability criteria, Phys. Rev. A 69 (2004) 022308], we introduce parallel definitions in the context of Gaussian states and using only properties of their covariance matrices, derive necessary and sufficient conditions for their complete extendability. It turns out that the complete extendability property is equivalent to the separability property of a bipartite Gaussian state. Following the proof of quantum de Finetti theorem as outlined in Hudson and Moody [Locally normal symmetric states and an analogue of de Finetti’s theorem, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 33(4) (1975/76) 343–351], we show that separability is equivalent to complete extendability for a state in a bipartite Hilbert space where at least one of which is of dimension greater than 2. This, in particular, extends the result of Fannes, Lewis, and Verbeure [Symmetric states of composite systems, Lett. Math. Phys. 15(3) (1988) 255–260] to the case of an infinite dimensional Hilbert space whose C* algebra of all bounded operators is not separable.

Dynamics of Operator-Weighted Shifts

2019 ◽  
Vol 29 (08) ◽  
pp. 1950110 ◽  
Author(s):  
Zongbin Yin ◽  
Yuming Chen ◽  
Qiaomin Xiang
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Weighted Shift ◽  
Necessary And Sufficient Conditions ◽  
Weighted Shifts ◽  
Diagonal Operators ◽  
Weight Sequence ◽  
Necessary And Sufficient

This paper investigates the dynamics of bilateral operator-weighted shifts on [Formula: see text] with a weight sequence of positive diagonal operators on a Hilbert space [Formula: see text]. Necessary and sufficient conditions for the bilateral weighted shifts to be hypercyclic (subspace-hypercyclic, frequently hypercyclic, Devaney chaotic, respectively) are provided. As a consequence, it is shown that for any [Formula: see text]-set [Formula: see text] of positive numbers which is bounded and bounded away from zero, there exists an invertible bilateral operator-weighted shift [Formula: see text] such that [Formula: see text]. Furthermore, the (hereditary) Cesàro-hypercyclicity of the bilateral weighted shifts is characterized.

scholarly journals Two characterisations of additive *-automorphisms of B(H)

1996 ◽  
Vol 53 (3) ◽  
pp. 391-400 ◽  
Author(s):  
Lajos Molnár
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Complex Hilbert Space ◽  
Necessary And Sufficient Conditions ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Preserver Problems ◽  
Necessary And Sufficient

Let H be a complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. In this paper we give two necessary and sufficient conditions for an additive bijection of B(H) to be a *-automorphism. Both of the results in the paper are related to the so-called preserver problems.

scholarly journals Necessary and sufficient conditions for spectral sets

1981 ◽  
Vol 24 (3) ◽  
pp. 349-355
Author(s):  
Takayuki Furuta ◽  
Muneo Chō
Keyword(s):  
Hilbert Space ◽  
Complex Plane ◽  
Sufficient Conditions ◽  
Complex Hilbert Space ◽  
Necessary And Sufficient Conditions ◽  
Spectral Sets ◽  
Closed Set ◽  
Spectral Set ◽  
Necessary And Sufficient

We shall show necessary and sufficient conditions for which closed set X in the complex plane is a spectral set of an operator T on a complex Hilbert space.

TO A TENSOR ANALYSIS OF THE SOLVABILITY OF THE PROBLEM OF REALIZATION OF A SECOND-ORDER BILINEAR SYSTEM WITH DELAY

2021 ◽  
Vol 2 ◽  
pp. 79-92
Author(s):  
Anatoly Lakeyev ◽  
◽  
Vyacheslav Rusanov ◽  
Andrey Banshchikov ◽  
◽  
...  
Keyword(s):  
Hilbert Space ◽  
Evolution Equations ◽  
Sufficient Conditions ◽  
Bilinear System ◽  
Second Order ◽  
Infinite Dimensional ◽  
Type Property ◽  
Machine Interface ◽  
Linear Differential ◽  
Necessary And Sufficient

The analytical conditions (necessary and sufficient) are defined for the solvability of the problem of differential realization of a continuous beam of controlled trajectory curves in the class of bilinear nonautonomous ordinary differential equations (with delay and without it) of the second order in a real separable Hilbert space. The problem under consideration belongs to the type of inverse problems for an additive combination of nonstationary linear and bilinear operators of evolution equations in an infinite-dimensional Hilbert space. The meta-language of this theory is the constructions of tensor products of Hilbert spaces, the structures of lattices with ortho-complementation, and the functional apparatus of the nonlinear Rayleigh-Ritz operator. It is shown that in the case of a finite bundle of trajectories, the presence of a sublinearity-type property of this operator allows us to obtain sufficient conditions for the existence of such realizations. Along the way, the topological-metric conditions for the continuity of the projectivization of the nonlinear Rayleigh-Ritz functional operator with the calculation of the fundamental group of its image are justified. The results obtained provide the motivation for the development of a qualitative theory of nonlinear structural identification of higher-order multi-linear differential models (e.g. for processes, induced by the «brain–machine» interface-platform of the type of Neuralink).

scholarly journals Subnormality of Powers of Multivariable Weighted Shifts

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Sang Hoon Lee ◽  
Woo Young Lee ◽  
Jasang Yoon
Keyword(s):  
Hilbert Space ◽  
Open Problem ◽  
Operator Theory ◽  
Sufficient Conditions ◽  
Weighted Shift ◽  
Necessary And Sufficient Conditions ◽  
Weighted Shifts ◽  
Hilbert Space Operators ◽  
Necessary And Sufficient

Given a pair T ≡ T 1 , T 2 of commuting subnormal Hilbert space operators, the Lifting Problem for Commuting Subnormals (LPCS) asks for necessary and sufficient conditions for the existence of a commuting pair N ≡ N 1 , N 2 of normal extensions of T 1 and T 2 ; in other words, T is a subnormal pair. The LPCS is a longstanding open problem in the operator theory. In this paper, we consider the LPCS of a class of powers of 2 -variable weighted shifts. Our main theorem states that if a “corner” of a 2-variable weighted shift T = W α , β ≔ T 1 , T 2 is subnormal, then T is subnormal if and only if a power T m , n ≔ T 1 m , T 2 n is subnormal for some m , n ≥ 1 . As a corollary, we have that if T is a 2-variable weighted shift having a tensor core or a diagonal core, then T is subnormal if and only if a power of T is subnormal.

scholarly journals Biquasitriangularity and spectral continuity

1985 ◽  
Vol 26 (2) ◽  
pp. 177-180 ◽  
Author(s):  
Ridgley Lange
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Invariant Subspaces ◽  
Extension Property ◽  
Point Of View ◽  
Single Valued Extension Property ◽  
Continuity Theorem ◽  
Spectral Continuity ◽  
Necessary And Sufficient ◽  
Continuity Of The Spectrum

In [6] Conway and Morrell characterized those operators on Hilbert space that are points of continuity of the spectrum. They also gave necessary and sufficient conditions that a biquasitriangular operator be a point of spectral continuity. Our point of view in this note is slightly different. Given a point T of spectral continuity, we ask what can then be inferred. Several of our results deal with invariant subspaces. We also give some conditions characterizing a biquasitriangular point of spectral continuity (Theorem 3). One of these is that the operator and its adjoint both have the single-valued extension property.

scholarly journals On an infinite dimensional linear-quadratic problem with fixed endpoints: The continuity question

2014 ◽  
Vol 24 (4) ◽  
pp. 723-733
Author(s):  
K.Maciej Przyłuski
Keyword(s):  
Minimum Energy ◽  
Sufficient Conditions ◽  
Control Problems ◽  
Linear Quadratic Control ◽  
Linear Quadratic ◽  
Minimum Norm Solution ◽  
Quadratic Problem ◽  
Infinite Dimensional ◽  
Necessary And Sufficient ◽  
Quadratic Control Problems

Abstract In a Hilbert space setting, necessary and sufficient conditions for the minimum norm solution u to the equation Su = Rz to be continuously dependent on z are given. These conditions are used to study the continuity of minimum energy and linear-quadratic control problems for infinite dimensional linear systems with fixed endpoints.

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