scholarly journals Wavelet bases for a unitary operator

1995 ◽  
Vol 38 (2) ◽  
pp. 233-260 ◽  
Author(s):  
S. L. Lee ◽  
H. H. Tan ◽  
W. S. Tang
Keyword(s):  
Hilbert Space ◽  
Riesz Basis ◽  
Unitary Operator ◽  
Linear Span ◽  
General Setting ◽  
Complex Hilbert Space ◽  
Riesz Bases ◽  
Duality Principle ◽  
Spline Wavelets ◽  
Necessary And Sufficient

Let T be a unitary operator on a complex Hilbert space ℋ, and X, Y be finite subsets of ℋ. We give a necessary and sufficient condition for TZ(X): {Tnx: n ∈ Z, x ∈ X} to be a Riesz basis of its closed linear span 〈TZ(X)〉. If TZ(X) and TZ(Y) are Riesz bases, and 〈TZ(X)〉⊂〈TZ(Y)〉, then X is extendable to X′ such that TZ(X′) is a Riesz basis of TZ(Y) The proof provides an algorithm for the construction of Riesz bases for the orthogonal complement of 〈TZ(X)〉 in 〈TZ(Y)〉. In the case X consists of a single B-spline, the algorithm gives a natural and quick construction of the spline wavelets of Chui and Wang [2, 3]. Further, the duality principle of Chui and Wang in [3] and [4] is put in the general setting of biorthogonal Riesz bases in Hilbert space.

On Self-Adjoint Factorization of Operators

1969 ◽  
Vol 21 ◽  
pp. 1421-1426 ◽  
Author(s):  
Heydar Radjavi
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Unitary Operator ◽  
Normal Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Normal Operators ◽  
Adjoint Operators

The main result of this paper is that every normal operator on an infinitedimensional (complex) Hilbert space ℋ is the product of four self-adjoint operators; our Theorem 4 is an actually stronger result. A large class of normal operators will be given which cannot be expressed as the product of three self-adjoint operators.This work was motivated by a well-known resul t of Halmos and Kakutani (3) that every unitary operator on ℋ is the product of four symmetries, i.e., operators that are self-adjoint and unitary.1. By “operator” we shall mean bounded linear operator. The space ℋ will be infinite-dimensional (separable or non-separable) unless otherwise specified. We shall denote the class of self-adjoint operators on ℋ by and that of symmetries by .

GENERALIZED p-FRAME IN SEPARABLE COMPLEX BANACH SPACES

2010 ◽  
Vol 08 (01) ◽  
pp. 133-148 ◽  
Author(s):  
XIANG-CHUN XIAO ◽  
YU-CAN ZHU ◽  
XIAO-MING ZENG
Keyword(s):  
Banach Space ◽  
Functional Analysis ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Riesz Basis ◽  
Complex Banach Space ◽  
Complex Hilbert Space ◽  
The Stability

The concept of g-frame and g-Riesz basis in a complex Hilbert space was introduced by Sun.18 In this paper, we generalize the g-frame and g-Riesz basis in a complex Hilbert space to a complex Banach space. Using operators theory and methods of functional analysis, we give some characterizations of a g-frame or a g-Riesz basis in a complex Banach space. We also give a result about the stability of g-frame in a complex Banach space.

Geometry on the Unit Ball of a Complex Hilbert Space

1978 ◽  
Vol 30 (01) ◽  
pp. 22-31 ◽  
Author(s):  
Kyong T. Hahn
Keyword(s):  
Hilbert Space ◽  
Unit Ball ◽  
Hyperbolic Geometry ◽  
Complex Hilbert Space ◽  
Open Unit ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Open Unit Ball ◽  
Similar Role ◽  
Necessary And Sufficient

Furnishing the open unit ball of a complex Hilbert space with the Carathéodory-differential metric, we construct a model which plays a similar role as that of the Poincaré model for the hyperbolic geometry. In this note we study the question whether or not through a point in the model not lying on a given line there exists a unique perpendicular, and give a necessary and sufficient condition for the existence of a unique perpendicular. This enables us to divide a triangle into two right triangles. Many trigonometric identities in a general triangle are easy consequences of various identities which hold on a right triangle.

scholarly journals Besselian g-frames and near g-Riesz bases

2011 ◽  
Vol 5 (2) ◽  
pp. 259-270 ◽  
Author(s):  
M.R. Abdollahpour ◽  
A. Najati
Keyword(s):  
Hilbert Space ◽  
Riesz Basis ◽  
Orthonormal Basis ◽  
Inner Product ◽  
Riesz Bases ◽  
Finite Dimensional

In this paper we introduce and study near g-Riesz basis, Besselian g-frames and unconditional g-frames. We show that a near g-Riesz basis is a Besselian g-frame and we conclude that under some conditions the kernel of associated synthesis operator for a near g-Riesz basis is finite dimensional. Finally, we show that a g-frame is a g-Riesz basis for a Hilbert space H if and only if there is an equivalent inner product on H, with respect to which it becomes an g-orthonormal basis for H.

scholarly journals The isometries of Hp(K)

1991 ◽  
Vol 50 (1) ◽  
pp. 23-33 ◽  
Author(s):  
Pei-Kee Lin
Keyword(s):  
Hilbert Space ◽  
Unit Disc ◽  
Unitary Operator ◽  
Complex Hilbert Space ◽  
Conformal Map ◽  
Image Position

AbstractLet 1 ≤ p < ∞, p ≠ 2 and let K be any complex Hilbert space. We prove that every isometry T of Hp(K) onto itself is of the form , where U ia a unitary operator on K and φ is a conformal map of the unit disc onto itself.

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.

Fidelity, sub-fidelity, super-fidelity and their preservers

2015 ◽  
Vol 13 (03) ◽  
pp. 1550027
Author(s):  
Li Wang ◽  
Jinchuan Hou ◽  
Kan He
Keyword(s):  
Hilbert Space ◽  
Unitary Operator ◽  
Convex Set ◽  
Quantum Systems ◽  
Complex Hilbert Space ◽  
Quantum States ◽  
Infinite Dimensional

Sub- and super-fidelity describe respectively the lower and super bound of fidelity of quantum states. In this paper, we obtain several properties of sub- and super-fidelity for both finite- and infinite-dimensional quantum systems. Furthermore, let H be a separable complex Hilbert space and ϕ : 𝒮(H) → 𝒮(H) a map, where 𝒮(H) denotes the convex set of all states on H. We show that, if dim H < ∞, or, if dim H = ∞ and ϕ is surjective, then the following statements are equivalent: (1) ϕ preserves the super-fidelity; (2) ϕ preserves the fidelity; (3) ϕ preserves the sub-fidelity; (4) there exists a unitary or an anti-unitary operator U on H such that ϕ(ρ) = UρU† for all ρ ∈ 𝒮(H).

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.

scholarly journals Extensions of dissipative operators with closable imaginary part

2021 ◽  
Vol 41 (3) ◽  
pp. 381-393
Author(s):  
Christoph Fischbacher
Keyword(s):  
Hilbert Space ◽  
Complex Hilbert Space ◽  
Dissipative Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Order Operator ◽  
Dissipative Operators ◽  
First Order ◽  
Necessary And Sufficient ◽  
Symmetric Operators

Given a dissipative operator \(A\) on a complex Hilbert space \(\mathcal{H}\) such that the quadratic form \(f \mapsto \text{Im}\langle f, Af \rangle\) is closable, we give a necessary and sufficient condition for an extension of \(A\) to still be dissipative. As applications, we describe all maximally accretive extensions of strictly positive symmetric operators and all maximally dissipative extensions of a highly singular first-order operator on the interval.

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