Operator-valued Analytic Functions Research Articles

Commuting row contractions with polynomial characteristic functions

A characteristic function is a special operator-valued analytic function defined on the open unit ball of $\mathbb{C}^n$ associated with an $n$-tuple of commuting row contraction on some Hilbert space. In this paper, we continue our study of the representations of $n$-tuples of commuting row contractions on Hilbert spaces, which have polynomial characteristic functions. Gleason's problem plays an important role in the representations of row contractions. We further complement the representations of our row contractions by proving theorems concerning factorizations of characteristic functions. We also emphasize the importance and the role of the noncommutative operator theory and noncommutative varieties to the classification problem of polynomial characteristic functions.

The generalized Birman–Schwinger principle

We prove a generalized Birman–Schwinger principle in the non-self-adjoint context. In particular, we provide a detailed discussion of geometric and algebraic multiplicities of eigenvalues of the basic operator of interest (e.g., a Schrödinger operator) and the associated Birman–Schwinger operator, and additionally offer a careful study of the associated Jordan chains of generalized eigenvectors of both operators. In the course of our analysis we also study algebraic and geometric multiplicities of zeros of strongly analytic operator-valued functions and the associated Jordan chains of generalized eigenvectors. We also relate algebraic multiplicities to the notion of the index of analytic operator-valued functions and derive a general Weinstein–Aronszajn formula for a pair of non-self-adjoint operators.

The Probabilistic Point of View on the Generalized Fractional Partial Differential Equations

Abstract This paper aims at unifying and clarifying the recent advances in the analysis of the fractional and generalized fractional Partial Differential Equations of Caputo and Riemann-Liouville type arising essentially from the probabilistic point of view. This point of view leads to the path integral representation for the solutions of these equations, which is seen to be stable with respect to the initial data and key parameters and is directly amenable to numeric calculations (Monte-Carlo simulation). In many cases these solutions can be compactly presented via the wide class of operator-valued analytic functions of the Mittag-Leffler type, which are proved to be expressed as the Laplace transforms of the exit times of monotone Markov processes.

A remark on analytic Fredholm alternative

We apply a recent result of Borichev-Golinskii-Kupin on the Blaschke-type conditions for zeros of analytic functions on the complex plane with a cut along the positive semi-axis to the problem of the eigenvalues distribution of the Fredholm-type analytic operator-valued functions.

Erratum to: On Factorizations of Analytic Operator-Valued Functions and Eigenvalue Multiplicity Questions

k=−N0 (z − z0)Kk(z0), 0 < |z − z0| < e0, for some N0 = N0(z0) ∈ N and some 0 < e0 = e0(z0) sufficiently small, with K−k(z0) ∈ F(K), 1 ≤ k ≤ N0(z0), Kk(z0) ∈ B(K), k ∈ N ∪ {0}, one should assume in addition to Hypothesis 2.5 that [IK −K0(z0)] ∈ Φ(K). No principal result in the original publication is affected by these corrections. We would like to thank Jussi Behrnd for kindly pointing the omission of [IK −K0(z0)] ∈ Φ(K) out to us.

On Factorizations of Analytic Operator-Valued Functions and Eigenvalue Multiplicity Questions

We study several natural multiplicity questions that arise in the context of the Birman–Schwinger principle applied to non-self-adjoint operators. In particular, we re-prove (and extend) a recent result by Latushkin and Sukhtyaev by employing a different technique based on factorizations of analytic operator-valued functions due to Howland. Factorizations of analytic operator-valued functions are of particular interest in themselves and again we re-derive Howland’s results and subsequently extend them. Considering algebraic multiplicities of finitely meromorphic operator-valued functions, we recall the notion of the index of a finitely meromorphic operator-valued function and use that to prove an analog of the well-known Weinstein–Aronszajn formula relating algebraic multiplicities of the underlying unperturbed and perturbed operators. Finally, we consider pairs of projections for which the difference belongs to the trace class and relate their Fredholm index to the index of the naturally underlying Birman–Schwinger operator.

The Schur Problem and Block Operator CMV Matrices

The block operator CMV matrices and their sub-matrices are applied to the description of all solutions to the Schur interpolation problem for contractive analytic operator-valued functions in the unit disk (the Schur class functions).

A remark on commutativity of the image of A-Valued analytic function

Let BL(ℍ) be a complex Banach algebra of all bounded linear operators acting on the Hilbert space ℍ. J. Globevnik and I. Vidav proved that when the range of an operator-valued analytic function with domain of definition D ⊂ ℂ consists of the normal operators, then f(D) is a commutative set in the algebra BL(ℍ). The paper strengthens this result for the topological algebras.

Convolution powers in the operator-valued framework

We consider the framework of an operator-valued noncommutative probability space over a unital C*-algebra B. We show how for a B-valued distribution \mu one can define convolution powers with respect to free additive convolution and with respect to Boolean convolution, where the exponent considered in the power is a suitably chosen linear map \eta from B to B, instead of being a non-negative real number. More precisely, the Boolean convolution power is defined whenever \eta is completely positive, while the free additive convolution power is defined whenever \eta - 1 is completely positive (where 1 stands for the identity map on B). In connection to these convolution powers we define an evolution semigroup related to the Boolean Bercovici-Pata bijection. We prove several properties of this semigroup, including its connection to the B-valued free Brownian motion. We also obtain two results on the operator-valued analytic function theory related to the free additive convolution powers with exponent \eta. One of the results concerns analytic subordination for B-valued Cauchy-Stieltjes transforms. The other gives a B-valued version of the inviscid Burgers equation, which is satisfied by the Cauchy-Stieltjes transform of a B-valued free Brownian motion.

Hankel Forms and Embedding Theorems in Weighted Dirichlet Spaces

We show that for a fixed operator-valued analytic function $g$ the boundedness of the bilinear (Hankel-type) form $(f,h)\to\int_\D\trace{g'^*fh'}(1-|z|^2)^\alpha \, \ud A$, defined on appropriate cartesian products of dual weighted Dirichlet spaces of Schatten class-valued functions, is equivalent to corresponding Carleson embedding estimates.

The Maximum Principle for Holomorphic Operator Functions

We show that if an operator-valued analytic function f of a complex variable attains its maximum modulus at z0, then the coefficients of the nonconstant terms in the power series expansion about z0 cannot be invertible, provided a complex uniform convexity condition holds. One application is that the norm of the resolvent of an operator on a complex uniformly convex space cannot have a local maximum.

Redheffer Representations and Relaxed Commutant Lifting

It is well known that the solutions of a (relaxed) commutant lifting problem can be described via a linear fractional representation of the Redheffer type. The coefficients of such Redheffer representations are analytic operator-valued functions defined on the unit disc D of the complex plane. In this paper we consider the converse question. Given a Redheffer representation, necessary and sufficient conditions on the coefficients are obtained guaranteeing the representation to appear in the description of the solutions to some relaxed commutant lifting problem. In addition, a result concerning a form of non-uniqueness appearing in the Redheffer representations under consideration and an harmonic maximal principle, generalizing a result of A. Biswas, are proved. The latter two results can be stated both on the relaxed commutant lifting as well as on on the Redheffer representation level.

Hardy spaces of operator-valued analytic functions

We are concerned with Hardy and BMO spaces of operator-valued functions analytic in the unit disk of C. In the case of the Hardy space, we involve the atomic decomposition since the usual argument in the scalar setting is not suitable. Several properties (the Garsia-norm equivalent theorem, Carleson measure, and so on) of BMOA spaces are extended to the operator-valued setting. Then, the operator-valued H-1-BMOA duality theorem is proved. Finally, by the H-1-BMOA duality we present the Lusin area integral and Littlewood-Paley g-function characterizations of the operator-valued analytic Hardy space.

Stable Ranks of Banach Algebras of Operator-Valued Analytic Functions

Let E be a separable infinite-dimensional Hilbert space, and let H(D; (E)) denote the algebra of all functions f:D (E) that are holomorphic. If is a subalgebra of H(D; (E)) , then using an algebraic result of Corach and Larotonda, we derive that under some conditions, the Bass stable rank of is infinite. In particular, we deduce that the Bass (and hence topological stable ranks) of the Hardy algebra H (D; (E)), the disk algebra A(D; (E)) and the Wiener algebra W+(D; (E)) are all infinite.

On CNC commuting contractive tuples

The characteristic function has been an important tool for studying completely non-unitary contractions on Hilbert spaces. In this note, we consider completely non-coisometric contractive tuples of commuting operators on a Hilbert space H. We show that the characteristic function, which is now an operator-valued analytic function on the open Euclidean unit ball in ℂn, is a complete unitary invariant for such a tuple. We prove that the characteristic function satisfies a natural transformation law under biholomorphic mappings of the unit ball. We also characterize all operator-valued analytic functions which arise as characteristic functions of pure commuting contractive tuples.

A characteristic operator function for the class of n-hypercontractions

We consider a class of bounded linear operators on Hilbert space called n-hypercontractions which relates naturally to adjoint shift operators on certain vector-valued standard weighted Bergman spaces on the unit disc. In the context of n-hypercontractions in the class C 0 ⋅ we introduce a counterpart to the so-called characteristic operator function for a contraction operator. This generalized characteristic operator function W n , T is an operator-valued analytic function in the unit disc whose values are operators between two Hilbert spaces of defect type. Using an operator-valued function of the form W n , T , we parametrize the wandering subspace for a general shift invariant subspace of the corresponding vector-valued standard weighted Bergman space. The operator-valued analytic function W n , T is shown to act as a contractive multiplier from the Hardy space into the associated standard weighted Bergman space.

Characteristic Function of a Pure Commuting Contractive Tuple

A theorem of Sz.-Nagy and Foias [9] shows that the characteristic function $\theta_T(z) = −T + zD_{T^*}(1_H - zT^*)^{-1}D_T$ of a completely non-unitary contraction T is a complete unitary invariant for T. In this note we extend this theorem to the case of a pure commuting contractive tuple using a natural generalization of the characteristic function to an operator-valued analytic function defined on the open unit ball of $C^n$. This function is related to the curvature invariant introduced by Arveson [3].

Nehari and Carathéodory-Fejér type extension results for operator-valued functions on groups

Let G G be a compact abelian group having the property that its character group Γ \Gamma is totally ordered by a semigroup P P . We prove that every operator-valued function k k on G G of the form k ( x ) = ∑ γ ∈ ( − P ) γ ( x ) k γ k(x)=\sum \limits _{\gamma \in (-P)}\gamma (x)k_{\gamma } , such that the Hankel operator H k H_k is bounded, has an essentially bounded extension K K with | | K | | ∞ = | | H k | | ||K||_{\infty }=||H_k|| . The proof is based on Arveson’s Extension Theorem for completely positive functions on C ∗ C^* -algebras. Among the corollaries we have a Carathéodory-Fejér type result for analytic operator-valued functions defined on such groups.

Analytic Subordination Consequences of Free Markovianity

Using simple operator-valued analytic function considerations several general analytic subordination results for a freely Markovian triple of operator algebras are derived from the analytic subordination result for generalized resolvents in the case of a free increment with amalgamation. This includes a generalization of Biane's subordination result in the case of a unitary free multiplicative increment.

Differential inequalities of operator-valued analytic functions

Some differential inequalities and integral inequalities for operator-valued analytic functions are investigated.