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.

Lp Behavior of the Eigenfunctions of the Invariant Laplacian

1993 ◽  
Vol 36 (4) ◽  
pp. 458-465
Author(s):  
E. G. Kwon
Keyword(s):  
Harmonic Functions ◽  
Maximum Modulus ◽  
Open Unit ◽  
Complex Numbers ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Maximum Modulus Principle ◽  
Open Unit Ball ◽  
Image Position ◽  
Necessary And Sufficient

AbstractLet be the invariant Laplacian on the open unit ball B of Cn and let Xλ denote the set of those f € C2(B) such that counterparts of some known results on X0, i.e. on M-harmonic functions, are investigated here. We distinguish those complex numbers λ for which the real parts of functions in Xλ belongs to Xλ. We distinguish those λ for which the Maximum Modulus Priniple remains true. A kind of weighted Maximum Modulus Principle is presented. As an application, setting α ≥ ½ and λ = 4n2α(α — 1), we obtain a necessary and sufficient condition for a function f in Xλ to be represented asfor some F ∊ LP(∂B).

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.

scholarly journals On approximating curves associated with nonexpansive mappings

2011 ◽  
Vol 27 (1) ◽  
pp. 142-147
Author(s):  
FRANCESCA VETRO ◽  
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Unit Ball ◽  
Nonexpansive Mappings ◽  
Complex Hilbert Space ◽  
Open Unit ◽  
Open Unit Ball ◽  
Holomorphic Map ◽  
Nonexpansive Map ◽  
The Hyperbolic Metric

Let X be a Banach space with metric d. Let T, N : X → X be a strict d-contraction and a d-nonexpansive map, respectively. In this paper we investigate the properties of the approximating curve associated with T and N. Moreover, following [3], we consider the approximating curve associated with a holomorphic map f : B → α B and a ρ-nonexpansive map M : B → B, where B is the open unit ball of a complex Hilbert space H, ρ is the hyperbolic metric defined on B and 0 ≤ α < 1. We give conditions on f and M for this curve to be injective, and we show that this curve is continuous.

The uniform exponential stability of a class of linear differential–difference equations in a Hilbert space

1981 ◽  
Vol 89 (3-4) ◽  
pp. 201-215 ◽  
Author(s):  
Richard Datko
Keyword(s):  
Hilbert Space ◽  
Phase Space ◽  
Exponential Stability ◽  
Difference Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Uniform Exponential Stability ◽  
Linear Differential ◽  
Necessary And Sufficient

SynopsisA necessary and sufficient condition is given for the uniform exponential stability of certain autonomous differential–difference equations whose phase space is a Hilbert space. It is shown that this property is preserved when the delays depend homogeneously on a nonnegative parameter.

A two-parameter eigenvalue problem involving complex potentials

1990 ◽  
Vol 116 (1-2) ◽  
pp. 177-191
Author(s):  
M. Faierman
Keyword(s):  
Hilbert Space ◽  
Eigenvalue Problem ◽  
Differential Equations ◽  
Complex Potentials ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Parameter System ◽  
Complete Set ◽  
Necessary And Sufficient ◽  
Two Parameter

SynopsisWe consider a two-parameter system of ordinary differential equations of the second order involving complex potentials and show that, unlike the case of real potentials, the eigenfunctions of the system do not necessarily form a complete set in the usual Hilbert space associated with the problem. We also give a necessary and sufficient condition for the eigenfunctions to be complete. Finally, we establish some results concerning the eigenvalues of the system.

Spectra of composition operators on algebras of analytic functions on Banach spaces

2009 ◽  
Vol 139 (1) ◽  
pp. 107-121 ◽  
Author(s):  
P. Galindo ◽  
T. W. Gamelin ◽  
Mikael Lindström
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Fixed Point ◽  
Unit Ball ◽  
Banach Spaces ◽  
Composition Operator ◽  
Composition Operators ◽  
Open Unit ◽  
Open Unit Ball ◽  
Analytic Symbol

Let E be a Banach space, with unit ball BE. We study the spectrum and the essential spectrum of a composition operator on H∞(BE) determined by an analytic symbol with a fixed point in BE. We relate the spectrum of the composition operator to that of the derivative of the symbol at the fixed point. We extend a theorem of Zheng to the context of analytic symbols on the open unit ball of a Hilbert space.

scholarly journals A–∞-interpolation in the ball

1998 ◽  
Vol 41 (2) ◽  
pp. 359-367 ◽  
Author(s):  
Xavier Massaneda
Keyword(s):  
Unit Ball ◽  
Polynomial Growth ◽  
Entire Functions ◽  
Holomorphic Functions ◽  
Weighted Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Derivatives Of

We give a necessary and sufficient condition for a sequence {ak}k in the unit ball of ℂn to be interpolating for the class A–∞ of holomorphic functions with polynomial growth. The condition, which goes along the lines of the ones given by Berenstein and Li for some weighted spaces of entire functions and by Amar for H∞ functions in the ball, is given in terms of the derivatives of m ≥ n functions F1, …,Fm ∈ A–∞ vanishing on {ak}k.

scholarly journals Binary Operations in the Unit Ball: A Differential Geometry Approach

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1178 ◽  
Author(s):  
Nikita E. Barabanov ◽  
Abraham A. Ungar
Keyword(s):  
Differential Geometry ◽  
Unit Ball ◽  
Hyperbolic Geometry ◽  
Binary Operation ◽  
Open Unit ◽  
Large Set ◽  
Open Unit Ball ◽  
Binary Operations ◽  
Möbius Addition ◽  
Metric Tensors

Within the framework of differential geometry, we study binary operations in the open, unit ball of the Euclidean n-space R n , n ∈ N , and discover the properties that qualify these operations to the title addition despite the fact that, in general, these binary operations are neither commutative nor associative. The binary operation of the Beltrami-Klein ball model of hyperbolic geometry, known as Einstein addition, and the binary operation of the Beltrami-Poincaré ball model of hyperbolic geometry, known as Möbius addition, determine corresponding metric tensors in the unit ball. For a variety of metric tensors, including these two, we show how binary operations can be recovered from metric tensors. We define corresponding scalar multiplications, which give rise to gyrovector spaces, and to norms in these spaces. We introduce a large set of binary operations that are algebraically equivalent to Einstein addition and satisfy a number of nice properties of this addition. For such operations we define sets of gyrolines and co-gyrolines. The sets of co-gyrolines are sets of geodesics of Riemannian manifolds with zero Gaussian curvatures. We also obtain a special binary operation in the ball, which is isomorphic to the Euclidean addition in the Euclidean n-space.

scholarly journals On the solution of Nevanlinna Pick problem with selfadjoint extensions of symmetric linear relations in Hilbert space

1997 ◽  
Vol 20 (3) ◽  
pp. 457-464
Author(s):  
A. A. El-Sabbagh
Keyword(s):  
Hilbert Space ◽  
Linear Relation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Resolvent Matrix ◽  
Linear Relations ◽  
Symmetric Linear Relation ◽  
Necessary And Sufficient ◽  
Selfadjoint Extensions ◽  
Pick Problem

The representation of Nevanlinna Pick Problem is well known, see [7], [8] and [11]. The aim of this paper is to find the necessary and sufficient condition for the solution of Nevanlinna Pick Problem and to show that there is a one-to-one correspondence between the solutions of the Nevanlinna Pick Problem and the minimal selfadjoint extensions of symmetric linear relation in Hilbert space. Finally, we define the resolvent matrix which gives the solutions of the Nevanlinna Pick Problem.

20.—Deficiency Indices of Polynomials in Symmetric Differential Expressions, II

1975 ◽  
Vol 73 ◽  
pp. 301-306 ◽  
Author(s):  
Anton Zettl
Keyword(s):  
Hilbert Space ◽  
Differential Expression ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Polynomial ◽  
Deficiency Indices ◽  
Linear Differential ◽  
Image Position ◽  
Necessary And Sufficient ◽  
The Relationship

SynopsisGiven a symmetric (formally self-adjoint) ordinary linear differential expression L which is regular on the interval [0, ∞) and has C∞ coefficients, we investigate the relationship between the deficiency indices of L and those of p(L), where p(x) is any real polynomial of degree k > 1. Previously we established the following inequalities: (a) For k even, say k = 2r, N+(p(L)), N−(p(L)) ≧ r[N+(L)+N−(L)] and (b) for k odd, say k = 2r+1where N+(M), N−(M) denote the deficiency indices of the symmetric expression M (or of the minimal operator associated with M in the Hilbert space L2(0, ∞)) corresponding to the upper and lower half-planes, respectively. Here we give a necessary and sufficient condition for equality to hold in the above inequalities.

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