scholarly journals Wavelet-type frames and wavelet-type bases

2005 ◽  
Vol 2005 (20) ◽  
pp. 3237-3245
Author(s):  
M. M. Shamooshaky
Keyword(s):  
Functional Analysis ◽  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Frame Theory ◽  
Theoretic Approach ◽  
Classical Literature ◽  
Wavelet Theory ◽  
Bounded Linear ◽  
Wide Range

The concepts of basis and frame are studied in the classical literature of functional analysis, Fourier analysis, and wavelet theory in a wide range. In this paper, we consider an operator-theoretic approach to discrete frame theory on a separable Hilbert space. For this purpose, we define a special type of frames and bases, called wavelet-type frames and wavelet-type bases, obtained by acting with a family of bounded linear operators on some vectors, and then investigate the elementary properties of these concepts.

Generators of Nest Algebras

1974 ◽  
Vol 26 (3) ◽  
pp. 565-575 ◽  
Author(s):  
W. E. Longstaff
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Nest Algebra ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Closed Algebra ◽  
Nest Algebras ◽  
Weakly Closed

A collection of subspaces of a Hilbert space is called a nest if it is totally ordered by inclusion. The set of all bounded linear operators leaving invariant each member of a given nest forms a weakly-closed algebra, called a nest algebra. Nest algebras were introduced by J. R. Ringrose in [9]. The present paper is concerned with generating nest algebras as weakly-closed algebras, and in particular with the following question which was first raised by H. Radjavi and P. Rosenthal in [8], viz: Is every nest algebra on a separable Hilbert space generated, as a weakly-closed algebra, by two operators? That the answer to this question is affirmative is proved by first reducing the problem using the main result of [8] and then by using a characterization of nests due to J. A. Erdos [2].

Representation of certain Linear Operators in Hilbert Space

1971 ◽  
Vol 23 (1) ◽  
pp. 132-150 ◽  
Author(s):  
Bernard Niel Harvey
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Separable Hilbert Space ◽  
Finite Dimension ◽  
Linear Operators ◽  
Inner Product ◽  
Indefinite Metric ◽  
Bounded Linear ◽  
Bilinear Functional

In this paper we represent certain linear operators in a space with indefinite metric. Such a space may be a pair (H, B), where H is a separable Hilbert space, B is a bilinear functional on H given by B(x, y) = [Jx, y], [, ] is the Hilbert inner product in H, and J is a bounded linear operator such that J = J* and J2 = I. If T is a linear operator in H, then ‖T‖ is the usual operator norm. The operator J above has two eigenspaces corresponding to the eigenvalues + 1 and –1.In case the eigenspace in which J induces a positive operator has finite dimension k, a general spectral theory is known and has been developed principally by Pontrjagin [25], Iohvidov and Kreĭn [13], Naĭmark [20], and others.

On Reflexivity of Algebras

1981 ◽  
Vol 33 (6) ◽  
pp. 1291-1308 ◽  
Author(s):  
Mehdi Radjabalipour
Keyword(s):  
Hilbert Space ◽  
Natural Number ◽  
Separable Hilbert Space ◽  
Invariant Subspaces ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Closed Algebra ◽  
Direct Sum ◽  
Weakly Closed

For each natural number n we define to be the class of all weakly closed algebras of (bounded linear) operators on a separable Hilbert space H such that the lattice of invariant subspaces of and (alg lat )(n) are the same. (If A is an operator, A(n) denotes the direct sum of n copies of A; if is a collection of operators,. Also, alg lat denotes the algebra of all operators leaving all invariant subspaces of invariant.) In the first section we show that . In Section 2 we prove that every weakly closed algebra containing a maximal abelian self adjoint algebra (m.a.s.a.) is , and that . It is also shown that certain algebras containing a m.a.s.a. are necessarily reflexive.

scholarly journals Characterizations of woven frames

2020 ◽  
Vol 18 (05) ◽  
pp. 2050033
Author(s):  
A. Bhandari ◽  
S. Mukherjee
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Universal Bounds

In a separable Hilbert space [Formula: see text], two frames [Formula: see text] and [Formula: see text] are said to be woven if there are constants [Formula: see text] so that for every [Formula: see text], [Formula: see text] forms a frame for [Formula: see text] with the universal bounds [Formula: see text]. This paper provides methods of constructing woven frames. In particular, bounded linear operators are used to create woven frames from a given frame. Several examples are discussed to validate the results. Moreover, the notion of woven frame sequences is introduced and characterized.

Approximation and Similarity Classification of Stably Finitely Strongly Irreducible Decomposable Operators

2010 ◽  
Vol 62 (2) ◽  
pp. 305-319
Author(s):  
He Hua ◽  
Dong Yunbai ◽  
Guo Xianzhou
Keyword(s):  
Hilbert Space ◽  
Jacobson Radical ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear ◽  
Similarity Classification ◽  
Strongly Irreducible ◽  
Decomposable Operators ◽  
Decomposable Operator

AbstractLet 𝓗 be a complex separable Hilbert space and ℒ(𝓗) denote the collection of bounded linear operators on 𝓗. In this paper, we show that for any operator A ∈ ℒ(𝓗), there exists a stably finitely (SI) decomposable operator A∈, such that ‖A−A∈‖ 𝓗 ∈ andA′(A∈)/ rad A′(A∈) is commutative, where rad A′(A∈) is the Jacobson radical of A′(A∈). Moreover, we give a similarity classification of the stably finitely decomposable operators that generalizes the result on similarity classification of Cowen–Douglas operators given by C. L. Jiang.

scholarly journals Approximation by partial isometries

1986 ◽  
Vol 29 (2) ◽  
pp. 255-261 ◽  
Author(s):  
Pei Yuan Wu
Keyword(s):  
Hilbert Space ◽  
Complex Plane ◽  
Closed Subset ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Unitary Operators ◽  
Nonempty Closed Subset ◽  
Partial Isometries ◽  
Bounded Linear ◽  
Normal Operators

Let B(H) be the algebra of bounded linear operators on a complex separable Hilbert space H. The problem of operator approximation is to determine how closely each operator T ∈B(H) can be approximated in the norm by operators in a subset L of B(H). This problem is initiated by P. R. Halmo [3] when heconsidered approximating operators by the positive ones. Since then, this problem has been attacked with various classes L: the class of normal operators whose spectrum is included in a fixed nonempty closed subset of the complex plane [4], the classes of unitary operators [6] and invertible operators [1]. The purpose of this paper is to study the approximation by partial isometries.

scholarly journals On the convexity of functions

Filomat ◽  
2019 ◽  
Vol 33 (12) ◽  
pp. 3773-3781
Author(s):  
Ata Abu-Asad ◽  
Omar Hirzallah
Keyword(s):  
Hilbert Space ◽  
Real Number ◽  
Positive Real Number ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Unitarily Invariant Norm ◽  
Positive Real ◽  
Bounded Linear ◽  
Invariant Norm ◽  
Increasing Function

Let A,B, and X be bounded linear operators on a separable Hilbert space such that A,B are positive, X ? ?I, for some positive real number ?, and ? ? [0,1]. Among other results, it is shown that if f(t) is an increasing function on [0,?) with f(0) = 0 such that f(?t) is convex, then ?|||f(?A + (1-?)B) + f(?|A-B|)|||?|||?f(A)X + (1-?)Xf (B)||| for every unitarily invariant norm, where ? = min (?,1-?). Applications of our results are given.

Similarity Classification of Cowen-Douglas Operators

2004 ◽  
Vol 56 (4) ◽  
pp. 742-775 ◽  
Author(s):  
Chunlan Jiang
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
List Type ◽  
Type Number ◽  
Bounded Linear ◽  
Similarity Classification ◽  
Strongly Irreducible ◽  
Connected Open Subset

AbstractLet ℋ be a complex separable Hilbert space and ℒ(ℋ) denote the collection of bounded linear operators on ℋ. An operator A in ℒ(ℋ) is said to be strongly irreducible, if , the commutant of A, has no non-trivial idempotent. An operator A in ℒ(ℋ) is said to be a Cowen-Douglas operator, if there exists Ω, a connected open subset of C, and n, a positive integer, such that(a)Ω ⊂ σ(A) = ﹛z ∈ C | A – z not invertible﹜;(b)ran(A – z) = ℋ, for z in Ω;(c)Vz∈Ω ker(A – z) = ℋ and(d)dim ker(A – z) = n for z in Ω.In the paper, we give a similarity classification of strongly irreducible Cowen-Douglas operators by using the K0-group of the commutant algebra as an invariant.

Commutators of Operators on Hilbert Space

1965 ◽  
Vol 17 ◽  
pp. 695-708 ◽  
Author(s):  
Arlen Brown ◽  
P. R. Halmos ◽  
Carl Pearcy
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Bilateral Shift

The purpose of this paper is to record some progress on the problem of determining which (bounded, linear) operators A on a separable Hilbert space H are commutators, in the sense that there exist bounded operators B and C on H satisfying A = BC — CB. It is thus natural to consider this paper as a continuation of the sequence (2; 3; 5). In §2 we show that many infinite diagonal matrices (with scalar entries) are commutators and that every weighted unilateral and bilateral shift is a commutator.

On the Calkin Algebra and the Covering Homotopy Property, II

1977 ◽  
Vol 29 (1) ◽  
pp. 210-215
Author(s):  
John B. Conway
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Compact Operators ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Calkin Algebra ◽  
Homotopy Property ◽  
The Ideal

For a separable Hilbert space is the algebra of bounded linear operators on is the ideal of compact operators, and Π is the natural map of onto the Calkin algebra .

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