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.

Perturbations of AF-Algebras

1979 ◽  
Vol 31 (5) ◽  
pp. 1012-1016 ◽  
Author(s):  
John Phillips ◽  
Iain Raeburn
Keyword(s):  
Hilbert Space ◽  
Unit Ball ◽  
Von Neumann Algebra ◽  
Unitary Operator ◽  
Affirmative Answer ◽  
Positive Answer ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Image Position ◽  
Af Algebras

Let A and B be C*-algebras acting on a Hilbert space H, and letwhere A1 is the unit ball in A and d(a, B1) denotes the distance of a from B1. We shall consider the following problem: if ‖A – B‖ is sufficiently small, does it follow that there is a unitary operator u such that uAu* = B?Such questions were first considered by Kadison and Kastler in [9], and have received considerable attention. In particular in the case where A is an approximately finite-dimensional (or hyperfinite) von Neumann algebra, the question has an affirmative answer (cf [3], [8], [12]). We shall show that in the case where A and B are approximately finite-dimensional C*-algebras (AF-algebras) the problem also has a positive answer.

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 Bloch Mappings in Complex Hilbert Space

1977 ◽  
Vol 29 (2) ◽  
pp. 299-306 ◽  
Author(s):  
Fletcher D. Wicker
Keyword(s):  
Hilbert Space ◽  
Analytic Functions ◽  
Unit Disc ◽  
Bloch Function ◽  
Complex Hilbert Space ◽  
Function Concept ◽  
Bounded Analytic Functions ◽  
Bloch Functions ◽  
Normal Functions ◽  
The Family

Anderson, Clunie and Pommerenke defined and studied the family of Bloch functions on the unit disc (see [1]). This family strictly contains the space of bounded analytic functions. However, all Bloch functions are normal and therefore enjoy the “nice” properties of normal functions. The importance of the Bloch function concept is the combination of their richness as a family and their “nice” behavior.

scholarly journals Spectral properties of holomorphic automorphism with fixed point

1986 ◽  
Vol 28 (1) ◽  
pp. 25-30 ◽  
Author(s):  
T. Mazur ◽  
M. Skwarczyński
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Spectral Properties ◽  
Unitary Operator ◽  
Biholomorphic Mapping ◽  
Holomorphic Automorphism ◽  
Image Position

The Hilbert space methods in the theory of biholomorphic mappings were applied and developed by S. Bergman [1, 2]. In this approach the central role is played by the Hilbert space L2H(D) consisting of all functions which are square integrable and holomorphic in a domain D ⊂ ℂN. A biholomorphic mapping φ:D ⃗ G induces the unitary mapping Uφ:L2H(G) ⃗ L2H(D) defined by the formulaHere ∂φ/∂z denotes the complex Jacobian of φ. The mapping Uϕ is useful, since it permits to replace a problem for D by a problem for its biholomorphic image G (see for example [11], [13]). When ϕ is an automorphism of D we obtain a unitary operator Uϕ on L2H(D).

Remarks Concerning Uniformly Bounded Operators on Hilbert Space

1972 ◽  
Vol 15 (2) ◽  
pp. 215-217 ◽  
Author(s):  
I. Istrǎțescu
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Selfadjoint Operator ◽  
Unitary Operator ◽  
Unitary Operators ◽  
Bounded Operators ◽  
Image Position ◽  
Uniformly Bounded

In [6] B. Sz.-Nagy has proved that every operator on a Hilbert space such that1is similar to a unitary operator.The following problem is an extension of this result: If T and S are two operators such that1.sup {‖Tn‖, ‖Sn‖}<∞ (n = 0, ±1, ±2,…)2.TS = STthen there exists a selfadjoint operator Q such that QTQ-1, QSQ-1 are unitary operators?Also, in [7] B. Sz.-Nagy has proved that every compact operator T such thatsup ‖Tn‖<∞ (n = 1, 2, 3,…)is similar to a contraction.

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 A characterization of spectral operators on Hilbert spaces

1982 ◽  
Vol 23 (1) ◽  
pp. 91-95 ◽  
Author(s):  
Ernst Albrecht
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Essential Spectrum ◽  
Hilbert Spaces ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Spectral Operator ◽  
Bounded Linear ◽  
Image Position

Let H be a complex Hilbert space and denote by B(H) the Banach algebra of all bounded linear operators on H. In [5; 6] J. Ph. Labrousse proved that every operator S∈B(H) which is spectral in the sense of N. Dunford (see [3]) is similar to a T∈B(H) with the following propertyConversely, he showed that given an operator S∈B(H) such that its essential spectrum (in the sense of [5; 6]) consists of at most one point and such that S is similar to a T∈B(H) with the property (1), then S is a spectral operator. This led him to the conjecture that an operator S∈B(H) is spectral if and only if it is similar to a T∈B(H) with property (1). The purpose of this note is to prove this conjecture in the case of operators which are decomposable in the sense of C. Foias (see [2]).

scholarly journals A Counterexample in the theory of derivations

1989 ◽  
Vol 31 (2) ◽  
pp. 161-163
Author(s):  
Feng Wenying ◽  
Ji Guoxing
Keyword(s):  
Hilbert Space ◽  
Trace Class ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Inner Product ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Schmidt Norm ◽  
Schmidt Operator ◽  
Image Position

Let B(H) be the algebra of all bounded linear operators on a separable, infinite dimensional complex Hilbert space H. Let C2 and C1 denote respectively, the Hilbert–Schmidt class and the trace class operators in B(H). It is known that C2 and C1 are two-sided*-ideals in B(H) and C2 is a Hilbert space with respect to the inner product(where tr denotes the trace). For any Hilbert–Schmidt operator X let ∥X∥2=(X, X)½ be the Hilbert-Schmidt norm of X.For fixed A ∈ B(H) let δA be the operator on B(H) defined byOperators of the form (1) are called inner derivations and they (as well as their restrictions have been extensively studied (for example [1–3]). In [1], Fuad Kittaneh proved the following result.

On triangle contractive operators in Hilbert spaces

1979 ◽  
Vol 85 (1) ◽  
pp. 17-20 ◽  
Author(s):  
Dang Dinh Ang ◽  
Le Hoan Hoa
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Hilbert Spaces ◽  
Complex Hilbert Space ◽  
Finite Dimensional ◽  
Fixed Line ◽  
Image Position

AbstractLet H be a finite dimensional real or complex Hilbert space. We denote by Λ(x, y, z) the area of the triangle with vertices x, y, z ∈ H. A map f: H → H is triangle contractive TC if 0 < α < 1 and for each x, y, z ∈ H eitherorandandWe prove that if f is TC either there is a fixed point w = f(w) or a fixed line L = ⊃ f(L) We characterize the f which are TC and continuous but have no fixed point.

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.

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