Dmitry Yakubovich

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• Ph. D.
• Professor (Associate) at Autonomous University of Madrid

We review Serguei Naboko’s criteria for similarity of an operator to unitary or selfadjoint one, their relation to his functional model, applications and some related results.

For an invertible linear operator T on a Hilbert space H, put α(T∗,T):=−T∗2T2+(1+r2)T∗T−r2I, where I stands for the identity operator on H and r∈(0,1); this expression comes from applying Agler's hereditary functional calculus to the polynomial α(t)=(1−t)(t−r2). We give a concrete unitarily equivalent functional model for operators satisfying α(T∗,...

We prove that in a large class of Banach spaces of analytic functions in the unit disc ⅅ an (unbounded) operator Af = G · f′ + g · f with G, g analytic in ⅅ generates a C0-semigroup of weighted composition operators if and only if it generates a C0-semigroup. Particular instances of such spaces are the classical Hardy spaces. Our result generalizes...

We prove that in a large class of Banach spaces of analytic functions in the unit disc $\mathbb{D}$ an (unbounded) operator $Af=G\cdot f'+g\cdot f$ with $G,\, g$ analytic in $\mathbb{D}$ generates a $C_0$-semigroup of weighted composition operators if and only if it generates a $C_0$-semigroup. Particular instances of such spaces are the classical...

For an invertible linear operator $T$ on a Hilbert space $H$, put \[ \alpha(T^*,T) := -T^{*2}T^2 + (1+r^2) T^* T - r^2 I, \] where $I$ stands for the identity operator on $H$ and $r\in (0,1)$; this expression comes from applying Agler's hereditary functional calculus to the polynomial $\alpha(t)=(1-t) (t-r^2)$. In the main part of this paper, we as...

We study the generalization of m-isometries and m-contractions (for positive integers m) to what we call a-isometries and a-contractions for positive real numbers a. We show that an operator satisfying a certain inequality in hereditary form is similar to a-contraction. This improvement of [9, Theorem I] is based on some Banach algebras techniques....

We discuss when an operator T, subject to a rather general inequality in hereditary form, admits a unitarily equivalent functional model of Agler type in the reproducing kernel Hilbert space associated to the inequality. The kernel need not be of Nevanlinna-Pick type. We define a defect operator D in our context and discuss the structure of the spe...

Finite rank perturbations T=N+K of a bounded normal operator N acting on a separable Hilbert space are studied thanks to a natural functional model of T; in its turn the functional model solely relies on a perturbation matrix/characteristic function previously defined by the second author. Function theoretic features of this perturbation matrix enc...

We study the generalization of $m$-isometries and $m$-contractions (for positive integers $m$) to what we call $a$-isometries and $a$-contractions for positive real numbers $a$. We show that any Hilbert space operator, satisfying an inequality of certain class (in hereditary form), is similar to $a$-contractions. This result is based on some Banach...

We discuss when an operator, subject to an inequality in heridatary form, admits a unitarily equivalent functional model of Agler type in the reproducing kernel Hilbert space associated to the inequality. To the contrary to the previous work, the kernel need not be of Nevanlinna-Pick type. We derive some consequences concerning the ergodic behavior...

Finite rank perturbations $T=N+K$ of a bounded normal operator $N$ on a separable Hilbert space are studied thanks to a natural functional model of $T$; in its turn the functional model solely relies on a perturbation matrix/ characteristic function previously defined by the second author. Function theoretic features of this perturbation matrix enc...

Let $T$ be a bounded linear operator on a Hilbert space $H$ such that \[ \alpha[T^*,T]:=\sum_{n=0}^\infty \alpha_n T^{*n}T^n\ge 0. \] where $\alpha(t)=\sum_{n=0}^\infty \alpha_n t^n$ is a suitable analytic function in the unit disc $\mathbb{D}$ with real coefficients. We prove that if $\alpha(t) = (1-t) \tilde{\alpha} (t)$, where $\tilde{\alpha}$ h...

We give some new criteria for a Hilbert space operator with spectrum on a smooth curve to be similar to a normal operator, in terms of pointwise and integral estimates of the resolvent. These results generalize criteria of Stampfli, Van Casteren and Naboko, and answers several questions posed by Stampfli in [48]. The main tools are from our recent...

Let $T$ be a bounded linear operator on a Hilbert space $H$ such that \[ \alpha[T^*,T]:=\sum_{n=0}^\infty \alpha_n T^{*n}T^n\ge 0. \] where $\alpha(t)=\sum_{n=0}^\infty \alpha_n t^n$ is a suitable analytic function in the unit disc $\mathbb{D}$ with real coefficients. We prove that if $\alpha(t) = (1-t) \tilde{\alpha} (t)$, where $\tilde{\alpha}$ h...

Avicou, Chalendar and Partington proved that an (unbounded) operator $(Af)=G\cdot f'$ on the classical Hardy space generates a $C_0$ semigroup of composition operators if and only if it generates a quasicontractive semigroup. Here we prove that if such an operator $A$ generates a $C_0$ semigroup, then it is automatically a semigroup of composition...

Avicou, Chalendar and Partington proved that an (unbounded) operator $(Af)=G\cdot f'$ on the classical Hardy space generates a $C_0$ semigroup of composition operators if and only if it generates a quasicontractive semigroup. Here we prove that if such an operator $A$ generates a $C_0$ semigroup, then it is automatically a semigroup of composition...

We give some new criteria for a Hilbert space operator with spectrum on a smooth curve to be similar to a normal operator, in terms of pointwise and integral estimates of the resolvent. These results generalize criteria of Stampfli, Van Casteren and Naboko, and answer several questions posed by Stampfli. The main tools are from our recent results o...

Given a Hilbert space operator $T$, the level sets of function $\Psi_T(z)=\|(T-z)^{-1}\|^{-1}$ determine the so-called pseudospectra of $T$. We set $\Psi_T$ to be zero on the spectrum of $T$. After giving some elementary properties of $\Psi_T$ (which, as it seems, were not noticed before), we apply them to the study of the approximation. We prove t...

Given a Hilbert space operator $T$, the level sets of function $\Psi_T(z)=\|(T-z)^{-1}\|^{-1}$ determine the so-called pseudospectra of $T$. We set $\Psi_T$ to be zero on the spectrum of $T$. After giving some elementary properties of $\Psi_T$ (which, as it seems, were not noticed before), we apply them to the study of the approximation. We prove t...

Let $\Phi$ be a family of functions analytic in some neighborhood of a complex domain $\Omega$, and let $T$ be a Hilbert space operator whose spectrum is contained in $\overline\Omega$. Our typical result shows that under some extra conditions, if the closed unit disc is complete $K'$-spectral for $\phi(T)$ for every $\phi\in \Phi$, then $\overline...

Suppose $\mathcal{A}$ is a compact normal operator on a Hilbert space $H$ with certain lacunarity condition on the spectrum (which means, in particular, that its eigenvalues go to zero exponentially fast), and let $\mathcal{L}$ be its rank one perturbation. We show that either infinitely many moment equalities hold or the linear span of root vector...

Suppose $\mathcal{A}$ is a compact normal operator on a Hilbert space $H$ with certain lacunarity condition on the spectrum (which means, in particular, that its eigenvalues go to zero exponentially fast), and let $\mathcal{L}$ be its rank one perturbation. We show that either infinitely many moment equalities hold or the linear span of root vector...

Given a complex domain $\Omega$ and analytic functions $\varphi_1,\ldots,\varphi_n : \Omega \to \mathbb{D}$, we give geometric conditions for $H^\infty(\Omega)$ to be generated by functions of the form $g \circ \varphi_k$, $g \in H^\infty(\mathbb{D})$. We apply these results to the extension of bounded functions on an analytic one-dimensional compl...

We introduce and study a class of operator tuples in complex Hilbert spaces, which we call spherical tuples. In particular, we characterize spherical multi-shifts, and more generally, multiplication tuples on RKHS. We further use these characterizations to describe various spectral parts including the Taylor spectrum. We also find a criterion for t...

Let $u(t)=-Fx(t)$ be the optimal control of the open-loop system $x'(t)=Ax(t)+Bu(t)$ in a linear quadratic optimization problem. By using different complex variable arguments, we give several lower and upper estimates of the exponential decay rate of the closed-loop system $x'(t)=(A-BF)x(t)$. Main attention is given to the case of a skew-Hermitian...

We study spectral properties of one-dimensional singular perturbations of an unbounded selfadjoint operator and give criteria for the possibility to remove the whole spectrum by a perturbation of this type. A counterpart of our results for the case of bounded operators provides a complete description of compact selfadjoint operators whose rank one...

The present article is a review of recent developments concerning the notion of F{\o}lner sequences both in operator theory and operator algebras. We also give a new direct proof that any essentially normal operator has an increasing F{\o}lner sequence $\{P_n\}$ of non-zero finite rank projections that strongly converges to 1. The proof is based on...

We survey recent results concerning the hereditary completeness of some special systems of functions and the spectral synthesis problem for a related class of linear operators. We present a solution of the spectral synthesis problem for systems of exponentials in $L^2(-\pi, \pi)$. Analogous results are obtained for the systems of reproducing kernel...

A (linear unbounded) operator $L$ is called a finite-dimensional singular perturbation of an operator $A$ if their graphs differ in a finite-dimensional space. We study spectral properties of a one-dimensional singular perturbation of an unbounded selfadjoint operator. Our approach is based on a functional model for this operator. We give criteria...

This article analyzes F\olner sequences of projections for bounded linear operators and their relationship to the class of finite operators introduced by Williams in the 70ies. We prove that each essentially hyponormal operator has a proper F\olner sequence (i.e. a F\olner sequence of projections strongly converging to 1). In particular, any quasin...

We prove that a sectorial operator admits an H ∞-functional calculus if and only if it has a functional model of Nagy–Foiaş type. Furthermore, we give a concrete formula for the characteristic function (in a generalized sense) of such an operator. More generally, this approach applies to any sectorial operator by passing to a different norm (the Mc...

We find a system of two polynomial equations in two unknowns, whose solution allows to give an explicit expression of the conformal representation of a simply connected three sheeted compact Riemann surface onto the extended complex plane. This function appears in the description of the ratio asymptotic of multiple orthogonal polynomials with respe...

Let $X$ be a Banach space. It is proved that an analogue of the Rubio de Francia square function estimate for partial sums of the Fourier series of $X$-valued functions holds true for all disjoint collections of subintervals of the set of integers of equal length and for all exponents $p$ greater or equal than 2 if and only if the space $X$ is a UM...

A functional model for nondissipative unbounded perturbations of an unbounded self-adjoint operator on a Hilbert space X is constructed. This model is analogous to the Nagy--Foias model of dissipative operators, but it is linearly similar and not unitarily equivalent to the operator. It is attached to a domain of parabolic type, instead of a half-p...

For a continuous complex-valued function g on the real line without zeros, several notions of a mean winding number are introduced. We give necessary conditions for a Toeplitz operator with matrix-valued symbol G to be semi-Fredholm in terms of mean winding numbers of det G. The matrix function G is assumed to be continuous on the real line, and no...

The aim of this paper is to inter-relate several algebraic and analytic objects, such as real-type algebraic curves, quadrature domains, functions on them and rational matrix functions with special properties, and some objects from operator theory, such as vector Toeplitz operators and subnormal operators. Our tools come from operator theory, but s...

In this paper we report some investigations on the problem of controlling isomerization for small poly- atomic non-rigid molecules, using the LiNC/LiCN system as an example. Two methods of control in the classical en- semble of LiNC/LiCN system are described and analyzed by performing computer simulations for the corresponding canonical ensemble. T...

The aim of this paper is to inter-relate several algebraic and analytic objects, such as real-type algebraic curves, quadrature domains, functions on them and rational matrix functions with special properties, and some objects from Operator Theory, such as vector Toeplitz operators and subnormal operators. Our tools come from operator theory, but s...

Starting from a continuous time linear time-invariant system, a linearly similar variant of the Nagy-Foias functional model of the main operator of the system is constructed. This model is formulated in terms of Hardy H 2 spaces in a semiplane. We give a connection between the construction of this model and the control theory (the input-state and t...

An analogue of the Szegö condition for density of analytic polynomials in a weighted Sobolev spaces Wk,p of the circle with general weights is given. This condition is always sufficient and is close to necessary. In particular, we prove that it is necessary and sufficient if all k + 1 weights are absolutely continuous and their densities are piecew...

Starting from a continuous time linear time-invariant system, a linearly similar variant of the Nagy{Foias model of the main operator of the system is constructed. A pole placement result, expressed in terms of invertibility of a certain Toeplitz operator, is obtained by applying this model. A numerical example of pole placement is given.

x0. Introduction In this paper we construct a new linearly similar functional model for linear operators and study its elementary properties. This model generalizes the Sz.-Nagy{Foia s model for C0-contractions and also forC0-dissipative operators. We shall not restrict ourselves to the disk or the half-plane: the model will be constructed in a fai...

A necessary condition for the exact controllability in neutral delay equations of general form is given in terms of uniform lower estimates for the minors of an extended transfer function. Some upper and lower estimates of the time of exact controllability in terms of the maximal delay and the dimensions of the state and control vectors are also ob...

In a series of papers [15], [16], [10], Putinar and Gustafsson connect the properties of a quadrature domain D with the properties of the hyponormal operator T with rank one self-commutator, whose principal function coincides with the characteristic function of this domain. Xia’s analytic model [19] of this hyponormal operator represents it as the...

This paper concerns pure subnormal operators with finite rank self-commutator, which we call subnormal operators of finite type. We analyze Xia's theory of these operators [21]-[23] and give its alternative exposition. Our exposition is based on the explicit use of a certain algebraic curve in C2, which we call the discriminant curve of a subnormal...

Xia proves in [9] that a pure subnormal operator S is completely determined by its self-commutator C = S*S - SS*, restricted to the closure M of its range and the operator ? = (S*|M)*. In [9], [10], [11] he constructs a model for S that involves this two operators and the so-called mosaic, which is a projection-valued function, analytic outside the...

Let T be a bounded linear operator in a separable Banach space X and let μ be a nonnegative measure in χ with compact support. A function mT,μ is considered that is defined μ-a.e. and has nonnegative integers or +∞ as values. This function is called the local multiplicity of T with respect to the measure μ. This function has some natural properties...

We consider a linear neutral functional equation of the form d dt Mx t = Lx t ; x(0) = c; x 0 = '; where M;L : L 2 Gamma [Gammah; 0]; C n Delta ! C n are unbounded linear operators of certain type and x t (`) = x(t+`), Gammah ` 0. Under certain conditions, the solution semigroup fT (t)g t0 is well-defined on the space Z = C n Theta L 2 Gamma [Gamma...

We consider a linear neutral functional equation of the form d dt Mx t = Lx t ; x(0) = c; x 0 = '; where M;L : L 2 Gamma [Gammah; 0]; C n Delta ! C n are unbounded linear operators of certain type and x t (`) = x(t+`), Gammah ` 0. Under certain conditions, the solution semigroup fT (t)g t0 is well-defined on the space Z = C n Theta L 2 Gamma [Gamma...

LetTbe a Banach space operator with empty point spectrum, whose essential spectrum lies on a finite system of (possibly intersecting) curves. Under certain conditions onT, dual analytic representations ofTas a kind of bundle shift and of T* as an “adjoint bundle shift” are constructed. A specialization of this scheme is given, which allows one to o...

In [6] (after Clancey's work [2]), Martin and Putinar introduced their two-dimensional functional model of a hyponormal operator, which reduces it to the multiplication by the independent variable in a space of distributions. Here we define another model which does (almost) the same for the adjoint operator. We also explain a close relation between...

In the paper one obtains the description of invariant subspaces of the multiplication operator in the Hardy-Smirnov space Ep (G), where G is afinitely connected domain with a piecewise C2-smooth boundary. For the case of an analytic “interior boundary” Γint of the domain G and p=2, a more precise description is given, generalizing the Hitt—Sarason...

Explicit formulas reducing a Toeplitz operator with rather general smooth symbol to diagonal form (in some sense), and explicit inverse formulas are constructed. The simultaneous completeness of eigenvectors of the operator and its adjoint, an analog of the von Neumann inequality and the existence of H-infinity-calculus are established. A triangula...

Let Ω⊂ℂ be a simply connected domain with a piecewise smooth boundary and assume that the function F is meromorphic in , does not have poles on δΩ, and the index of each point λ∈ℂ\F(δΩ) with respect to the curve F(δΩ) is nonnegative (at the positive traversal of the curve δΩ). Under these assumptions, for a class of Banach spaces (including the Har...

We consider a Toeplitz operator TF whose symbol F is continuous on the unit circle, analytic in the unit disc except for a finite number of poles and has non-negative winding number with respect to any point in C .It is shown that TF is similar to the multiplication operator by the projection II on a certain function space H F2(б*) on the so-called...

Let s be a weighted shift operator in lp, p ε[1+∞)∶s(b0, b1, ...)=(0, λ0, b0, λ1, b1, ...). One proves its unicellularity under the condition ¦λi¦ ↓ 0 and also under some weaker conditions. One obtains also unicellularity conditions for weighted shift operators in Banach spaces cf numerical sequences. One gives a new proof of the following theorem...

The conversion of a given polynomial rectangular matrix to a square one is considered, which ensures that the determinant of the latter is equal to a given one. The necessary and sufficient conditions for solubility of this problem are given, together with an algorithm for constructing the square matrix. The results can be used in control applicati...