Operator-valued Analytic Functions Research Articles

Noncommutative interpolation and Poisson transforms

General results of interpolation (e.g., Nevanlinna-Pick) by elements in the noncommutative analytic Toeplitz algebraF ∞ (resp., noncommutative disc algebraA n) with consequences to the interpolation by bounded operator-valued analytic functions in the unit ball of ℂn are obtained. Noncommutative Poisson transforms are used to provide new von Neumann type inequalities. Completely isometric representations of the quotient algebraF ∞/J on Hilbert spaces whereJ is anyw *-closed, 2-sided ideal ofF ∞, are obtained and used to construct aw *-continuous,F ∞/J-functional calculus associated to row contractionsT=[T 1,…,T n] whenf(T1, …, Tn)=0 for anyf∈J. Other properties of the dual algebraF ∞/J are considered.

Operator-Valued Typically Real Functions Induced by a Contraction

Let H be a complex Hilbert space and let L(H) denote the algebra of all bounded linear operators on H. LetF(z)=∑∞n=0znAnbe an operator-valued analytic function whose coefficients are bounded operators in L(H) forzin the open unit disc D and the series is convergent in the strong operator topology. In this paper, we discuss operator-valued typically real functionsF(z)=∑∞n=0znAnwhich generalize complex-valued typically real functions. We characterize operator-valued typically real functions and study such functionsF(z) induced by a contraction on H. In addition, we considerm×n-tuple operator-valued typically real functions and positively real functions. Finally, we charaterize operator-valued typically real functions in the finite-dimensional cases.

An algebraic multiplicity theory for analytic operator-valued functions

An algebraic multiplicity theory for analytic operator-valued functions

Perturbation of Invariant Subspaces

The questions of how perturbation of an operator affects its invariant subspaces have raised a substantial amount of interest, especially in the Ž w x . finite dimensions see, e.g., 5, 10]12 and the references given there . It turns out that linear perturbation of the operator gives rise to an analytic perturbation of the projection on its invariant subspace under some reasonable conditions. Although the question of estimating the norms of the terms of the Taylor series’ expansion of this projection-valued function does have some applications in numerical analysis, it seems to be studied w x less frequently. For instance, in 12 the matrix algebra approach combined with complex analysis in several variables was used to get existence and analyticity of the perturbed projection. The terms were obtained as well, using specially developed formulas for their direct computation. However, we find it hard to see that these estimates are good enough to ensure the convergence of this series, let alone to give an estimate of its radius of convergence. In this article we propose a different approach to the subject which achieves the two goals at one stroke and, we believe, with better results. Namely, we study the problem in terms of operator-valued analytic functions, thus reducing the problem to solving an algebraic Riccati-type equation. In order to solve it we extend in Sections 2 and 3 somewhat the Ž usual methods to our case see the comments about the meaning of . ‘‘usual’’ in the last paragraph of the introduction . The main result, given in Section 4, is the estimates of the radius of convergence and of the terms. The estimates are clearly good enough to yield an estimated radius of

The universal multiplicity theory for analytic operator-valued functions

An analytic operator-valued function A is an analytic map A: D → L(E, E), where D = D(A) is an open subset of the complex plane C and E = E(A) is a complex Banach space. For such a function A the singular set σ(A) of A is defined as the set of points z ∈ D such that A(z) is not invertible. It is a relatively closed subset of D.

Principal part of resolvent and factorization of an increasing nonanalytic operator-function

Let a selfadjoint operator-valued functionL(λ) be given on the interval [a,b] such thatL(a)≪0,L(b)≫0,L′(λ)≫0 (a≤λ≤b), andL″(λ) has a certain smoothness (for instance, it satisfies Holder's condition). It turns out that the spectral theory of the operator-valued functionL(λ) can be reduced to the spectral theory of one operatorZ, the spectrum of which lies on (a, b) and which is similar to a selfadjoint operator. In particular, the factorization takes place:L(λ)=M(λ)(λI−Z), where the operator-valued functionM(λ) is invertible on [a, b]. Earlier similar results were known only for analytic operator-valued functions. The authors had to use new methods for the proof of the described theorem. The key moment is the decomposition ofL−1(λ) into the sume of its principal and regular parts.

Operator-valued analytic functions of constant norm

Operator-valued analytic functions of constant norm

Uniqueness theorems for analytic operator-valued functions with nonnegative imaginary part

Uniqueness theorems for analytic operator-valued functions with nonnegative imaginary part

The angular derivative of an operator-valued analytic function

The angular derivative of an operator-valued analytic function

Commutative ranges of analytic functions in Banach algebras

Let A A denote a complex unital Banach algebra with Hermitian elements ( A ) (A) . We show that if F F is an analytic function from a connected open set D D into A A such that F ( z ) F(z) is normal ( F ( z ) = u ( z ) + i υ ( z ) (F(z) = u(z) + i\upsilon (z) , where u ( z ) u(z) , υ ( z ) ∈ H ( A ) \upsilon (z) \in H(A) and u ( z ) υ ( z ) = υ ( z ) u ( z ) ) u(z)\upsilon (z) = \upsilon (z)u(z)) for each z ∈ D z \in D , then F ( z ) F ( w ) = F ( w ) F ( z ) F(z)F(w) = F(w)F(z) for all w w , z ∈ D z \in D . This generalizes a theorem of Globevnik and Vidav concerning operator-valued analytic functions. As a corollary, it follows that an essentially normal-valued analytic function has an essentially commutative range.

Commutative Ranges of Analytic Functions in Banach Algebras

Let $A$ denote a complex unital Banach algebra with Hermitian elements $(A)$. We show that if $F$ is an analytic function from a connected open set $D$ into $A$ such that $F(z)$ is normal $(F(z) = u(z) + i\upsilon (z)$, where $u(z)$, $\upsilon (z) \in H(A)$ and $u(z)\upsilon (z) = \upsilon (z)u(z))$ for each $z \in D$, then $F(z)F(w) = F(w)F(z)$ for all $w$, $z \in D$. This generalizes a theorem of Globevnik and Vidav concerning operator-valued analytic functions. As a corollary, it follows that an essentially normal-valued analytic function has an essentially commutative range.

On operator-valued analytic functions with positive real part whose logarithm belongs to a $C_{p}$ class

On operator-valued analytic functions with positive real part whose logarithm belongs to a $C_{p}$ class

Inverse eigenvalue problem for a finite multiplying slab

Defines a general class of problem that has been termed the inverse eigenvalue problem. Basically similar problems have already been studied as isolated and specific examples in the analysis of time eigenvalues appearing in neutron transport theory. In this work, however, the authors present a general unified method for their treatment using functional analytic methods. Specifically, the critical slab problem has been analysed as an example of such an inverse eigenvalue problem of a Fredholm integral equation using the theory of perturbation of a class of positive, analytic operator-valued functions in Banach space. Numerical calculations of the critical thickness are given. These results are encouraging, considering the simplicity of the method, which does not involve an explicit solution of the Fredholm equation.

A Boundary Value Criterion for Semipassive Hilbert Ports

Necessary and sufficient conditions on the boundary values of an operator-valued analytic function Y defined on the open right half-plane are given which ensure that Y is the Laplace transform of the unit-impulse response of a linear time-invariant causal semipassive Hilbert port. Similar results are also obtained for the more restricted case where the Hilbert port is linear time-invariant and passive.

On Hankel Operator Ranges, Meromorphic Pseudo-Continuations and Factorization of Operator-Valued Analytic Functions†

Necessary and sufficient conditions for the existence of meromorphic pseudo-continuations of bounded type for bounded operator-valued analytic functions in the unit disc are given in terms of the ranges of the induced Hankel operators. Related factorizations of the functions are obtained.

On the representation and approximation of a class of operator-valued analytic functions

1. Main results. Let H be a separable Hilbert space, and a, b9 real and finite. Suppose that T(z) is, for each z O [a, b], a bounded operator-valued analytic function which satisfies the condition (R) For each (j> G H and z O [a, b], (r(z)0, 0) maps the upper halfplane into itself and the lower half-plane into itself We then call T(z) an R-operator, and the class of all /?-operators with the cut [a, b] will be called W[a, b]. The class (jj[a, b] generalizes to operators of the class R[a, b] of R-functions, those analytic functions on the cut complex plane which preserve upper and lower half-planes. For any f(z) G R[a, b], there is a nonnegative number a, a real number |8, and a positive measure JU with support on [a, b], such that

Continued fractions of operator-valued analytic functions

The subject of this work is the convergence of infinite continued fractions whose coefficients are analytic functions which take their values in the space of bounded linear operators on a complex Hilbert space. By appropriately combining a prior result of Fair with Vitali's theorem, we show that convergence under the uniform operator topology occurs on certain angular regions of the complex plane whenever the operator coefficients are commutative, invertible, and satisfy certain conditions on their numerical ranges.

On operator-valued analytic functions with constant norm

Let X be a complex Banach space and D a domain in the complex plane. Let f: D → X be an analytic function such that ∥ f(ζ)∥ is constant as ζ ϵ D . If X is the complex plane, then by the classical maximum modulus theorem f;(ζ) itself is constant on D . This is not the case in general. In the paper we study the norm-constant analytic functions whose values are bounded linear operators over an uniformly convex complex Banach space or, in particular, over a complex Hilbert space.

Eigenfunctions of Operator-Valued Analytic Functions

This paper is a sequel to [2], whose primary purposes are to clarify and generalize the concept introduced there of an eigenfunction of an inner function, and to answer questions raised there concerning the equivalence of several possible forms of the definition. A new definition, proposed here, leads to a complete characterization of the eigenfunctions of Potapov inner functions of normal operators, and the result is more satisfactory than [2, Theorem 3.4], although the latter is used strongly in the proof.Let be an inner function in the sense of Lax; i.e., is almost everywhere (a.e.) a unitary operator on a separable Hilbert space and belongs weakly to the Hardy class H2. An analytic function q (which will have to be a scalar inner function) was defined to be an eigenfunction of if the set of z in the disk {z: |z| ≦ 1} for which is invertible is a set of linear measure 0 on the circle {z: |z| = 1}.

Summability Methods in Perturbation Theory

The Mittag-Leffler summability method is applied to operator-valued analytic functions and a corresponding procedure for perturbation theory is derived, which has a bigger region of convergence. This region is explicitly described.