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).

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.

Positive Definite Sequence of Operators and a Fixed Point Theorem

1972 ◽  
Vol 15 (2) ◽  
pp. 295-295
Author(s):  
A. T. Dash
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Positive Definite ◽  
Definite Sequence ◽  
Positive Definite Sequence ◽  
Image Position

The purpose of this note is to prove the following:Theorem. Let {An} be a positive definite sequence of operators on a Hilbert space H with A0=1. If A1f=f for some f in H, then Anf=f for all n.Note that a bilateral sequence of operators {An:n = 0, ±1, ±2,…} on H is positive definite iffor every finitely nonzero sequence {fn} of vectors in H [1].

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.

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.

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.

On a Family of Generalized Numerical Ranges

1974 ◽  
Vol 26 (3) ◽  
pp. 678-685
Author(s):  
C.-S. Lin
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Banach Algebra ◽  
Orthogonal Projection ◽  
Bounded Linear Operator ◽  
Unitary Operator ◽  
Linear Operators ◽  
Minkowski Distance ◽  
Bounded Linear ◽  
Image Position

Throughout this note, an operator will always mean a bounded linear operator acting on a Hilbert space X into itself, unless otherwise stated. The class Cρ (0 < ρ < ∞ ) of operators, considered by Sz.-Nagy and Foiaş [5], is defined as follows: An operator T is in Cρ if Tnx = pPUnx for all x ∊ X, n = 1, 2, . . . , where U is a unitary operator on some Hilbert space Y containing X as a subspace, and P is the orthogonal projection of Y onto X. In [2] Holbrook defined the operator radii wρ(·) (0 < ρ ≦ ∞ ) as the generalized Minkowski distance functionals on the Banach algebra of bounded linear operators on X, i.e.,and w∞(T) = r(T), the spectral radius of T [2, Theorem 5.1].

Spectral Properties for Invertible Measure Preserving Transformations

1973 ◽  
Vol 25 (4) ◽  
pp. 806-811 ◽  
Author(s):  
Jean-Marc Belley
Keyword(s):  
Spectral Properties ◽  
Unitary Operator ◽  
Unit Interval ◽  
Complex Conjugation ◽  
Integrable Functions ◽  
Measure Preserving Transformation ◽  
Image Position ◽  
Square Integrable Functions ◽  
Measure Preserving Transformations

An invertible measure preserving transformation T on the unit interval I generates a unitary operator U on the space L2(I) of Lebesque square integrable functions given by (Uf)(x) = f(Tx) for all f in L2(I) and x in I. By definitionfor all f , g in L2(I), the bar denoting complex conjugation.

Global attracting sets of neutral stochastic functional integro-differential equations driven by a fractional Brownian motion

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
A. Bakka ◽  
S. Hajji ◽  
D. Kiouach
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Hurst Parameter ◽  
Fixed Point Principle ◽  
Stochastic Functional

Abstract By means of the Banach fixed point principle, we establish some sufficient conditions ensuring the existence of the global attracting sets of neutral stochastic functional integrodifferential equations with finite delay driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} in a Hilbert space.

scholarly journals Local Spectral Properties Under Conjugations

2021 ◽  
Vol 18 (3) ◽  
Author(s):  
Pietro Aiena ◽  
Fabio Burderi ◽  
Salvatore Triolo
Keyword(s):  
Hilbert Space ◽  
Spectral Properties ◽  
Operator Matrices ◽  
Weyl Type Theorems ◽  
Triangular Operator ◽  
Analytic Core ◽  
Complex Symmetric ◽  
Symmetric Operators ◽  
The Relationship

AbstractIn this paper, we study some local spectral properties of operators having form JTJ, where J is a conjugation on a Hilbert space H and $$T\in L(H)$$ T ∈ L ( H ) . We also study the relationship between the quasi-nilpotent part of the adjoint $$T^*$$ T ∗ and the analytic core K(T) in the case of decomposable complex symmetric operators. In the last part we consider Weyl type theorems for triangular operator matrices for which one of the entries has form JTJ, or has form $$JT^*J$$ J T ∗ J . The theory is exemplified in some concrete cases.

Factorization of an adjontable Markov operator

2019 ◽  
Vol 22 (02) ◽  
pp. 1950013
Author(s):  
Carlo Pandiscia
Keyword(s):  
Hilbert Space ◽  
Matrix Algebra ◽  
Unitary Operator ◽  
Markov Operator ◽  
Factorization Property ◽  
Markov Operators ◽  
Dilation Theory ◽  
Probability Spaces

In this work, we propose a method to investigate the factorization property of a adjontable Markov operator between two algebraic probability spaces without using the dilation theory. Assuming the existence of an anti-unitary operator on Hilbert space related to Stinespring representations of our Markov operator, which satisfy some particular modular relations, we prove that it admits a factorization. The method is tested on the two typologies of maps which we know admits a factorization, the Markov operators between commutative probability spaces and adjontable homomorphism. Subsequently, we apply these methods to particular adjontable Markov operator between matrix algebra which fixes the diagonal.

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