Valentin Matache

Composition operators on Hardy spaces of a half-plane

We consider composition operators on Hardy spaces of a halfplane. We mainly study boundedness and compactness. We prove that on these spaces there are no compact composition operators.

Numerical Ranges of Composition Operators

Abstract Composition operators on the Hilbert Hardy space of the unit disk are considered. The shape of their numerical range is determined in the case when the symbol of the composition operator is a monomial or an inner function fixing 0. Several results on the numerical range of composition operators of arbitrary symbol are obtained. It is proved that 1 is an extreme boundary point if and only if 0 is a fixed point of the symbol. If 0 is not a fixed point of the symbol, 1 is shown to be interior to the numerical range. Some composition operators whose symbol fixes 0 and has infinity norm less than 1 have closed numerical ranges in the shape of a cone-like figure, i.e., a closed convex region with a corner at 1, 0 in its interior, and no other corners. Compact composition operators induced by a univalent symbol whose fixed point is not 0 have numerical ranges without corners, except possibly a corner at 0.

Invertible and normal composition operators on the Hilbert Hardy space of a half–plane

Abstract Operators on function spaces of form Cɸf = f ∘ ɸ, where ɸ is a fixed map are called composition operators with symbol ɸ. We study such operators acting on the Hilbert Hardy space over the right half-plane and characterize the situations when they are invertible, Fredholm, unitary, and Hermitian. We determine the normal composition operators with inner, respectively with Möbius symbol. In select cases, we calculate their spectra, essential spectra, and numerical ranges.

Operator self-similar processes on Banach spaces

Operator self-similar (OSS) stochastic processes on arbitrary Banach spaces are considered. If the family of expectations of such a process is a spanning subset of the space, it is proved that the scaling family of operators of the process under consideration is a uniquely determined multiplicative group of operators. If the expectation-function of the process is continuous, it is proved that the expectations of the process have power-growth with exponent greater than or equal to 0, that is, their norm is less than a nonnegative constant times such a power-function, provided that the linear space spanned by the expectations has category 2 (in the sense of Baire) in its closure. It is shown that OSS processes whose expectation-function is differentiable on an interval (s0,∞), for some s0≥1, have a unique scaling family of operators of the form {sH:s>0}, if the expectations of the process span a dense linear subspace of category 2. The existence of a scaling family of the form {sH:s>0} is proved for proper Hilbert space OSS processes with an Abelian scaling family of positive operators.

Hilbert Spaces Induced by Toeplitz Covariance Kernels

We consider the reproducing kernel Hilbert space H μ induced by a kernel which is obtained using the Fourier-Stieltjes transform of a regular, positive, finite Borel measure μ on a locally compact abelian topological group Γ. Denote by G the dual of Γ. We determine H μ as a certain subspace of the space C o(G) of all continuous function on G vanishing at infinity. Our main application is calculating the reproducing kernel Hilbert spaces induced by the Toeplitz covariance kernels of some well-known stochastic processes.

On the sensitivity to noise of a Boolean function

In this paper we generate upper and lower bounds for the sensitivity to noise of a Boolean function using relaxed assumptions on input choices and noise. The robustness of a Boolean network to noisy inputs is related to the average sensitivity of that function. The average sensitivity measures how sensitive to changes in the inputs the output of the function is. The average sensitivity of Boolean functions can indicate whether a specific random Boolean network constructed from those functions is ordered, chaotic, or in critical phase. We give an exact formula relating the sensitivity to noise and the average sensitivity of a Boolean function. The analytic approach is supplemented by numerical results that illustrate the overall behavior of the sensitivities as various Boolean functions are considered. It is observed that, for certain parameter combinations, the upper estimates in this paper are sharper than other estimates in the literature and that the lower estimates are very close to the actual values ...

Problems on weighted and unweighted composition operators

This paper contains a collection of open problems related to composition operators and weighted composition operators acting on spaces of analytic functions, predominantly on the Hilbert Hardy space H2 over the open unit disc. Some are related to the invariant subspaces of composition operators, some to the spectra and numerical ranges of such operators, others are related to the connection of certain weighted composition on H2 and the unweighted composition operators on Hardy–Smirnov spaces, or the connection of composition operators with asymptotically Toeplitz operators. The problems raised are open to the knowledge of this author, and interesting, in his opinion.

Logical Reduction of Biological Networks to Their Most Determinative Components.

Boolean networks have been widely used as models for gene regulatory networks, signal transduction networks, or neural networks, among many others. One of the main difficulties in analyzing the dynamics of a Boolean network and its sensitivity to perturbations or mutations is the fact that it grows exponentially with the number of nodes. Therefore, various approaches for simplifying the computations and reducing the network to a subset of relevant nodes have been proposed in the past few years. We consider a recently introduced method for reducing a Boolean network to its most determinative nodes that yield the highest information gain. The determinative power of a node is obtained by a summation of all mutual information quantities over all nodes having the chosen node as a common input, thus representing a measure of information gain obtained by the knowledge of the node under consideration. The determinative power of nodes has been considered in the literature under the assumption that the inputs are independent in which case one can use the Bahadur orthonormal basis. In this article, we relax that assumption and use a standard orthonormal basis instead. We use techniques of Hilbert space operators and harmonic analysis to generate formulas for the sensitivity to perturbations of nodes, quantified by the notions of influence, average sensitivity, and strength. Since we work on finite-dimensional spaces, our formulas and estimates can be and are formulated in plain matrix algebra terminology. We analyze the determinative power of nodes for a Boolean model of a signal transduction network of a generic fibroblast cell. We also show the similarities and differences induced by the alternative complete orthonormal basis used. Among the similarities, we mention the fact that the knowledge of the states of the most determinative nodes reduces the entropy or uncertainty of the overall network significantly. In a special case, we obtain a stronger result than in previous works, showing that a large information gain from a set of input nodes generates increased sensitivity to perturbations of those inputs.

QUEUEING SYSTEMS FOR MULTIPLE FBM-BASED TRAFFIC MODELS

A multiple fractional Brownian motion (FBM)-based traffic model is considered. Various lower bounds for the overflow probability of the associated queueing system are obtained. Based on a probabilistic bound for the busy period of an ATM queueing system associated with a multiple FBM-based input traffic, a minimal dynamic buffer allocation function (DBAF) is obtained and a DBAF-allocation algorithm is designed. The purpose is to create an upper bound for the queueing system associated with the traffic. This upper bound, called a DBAF, is a function of time, dynamically bouncing with the traffic. An envelope process associated with the multiple FBM-based traffic model is introduced and used to estimate the queue size of the queueing system associated with that traffic model.

When is the numerical range of a nilpotent matrix circular

The problem formulated in the title is investigated. The case of nilpotent matrices of size at most 4 allows a unitary treatment. The numerical range of a nilpotent matrix M of size at most 4 is circular if and only if the traces trM^*M^2 and trM^*M^3 are null. The situation becomes more complicated as soon as the size is 5. The conditions under which a 5x5 nilpotent matrix has circular numerical range are thoroughly discussed.