arXiv subjects
Publications and source records attributed to Dmitry Yakubovich.
Let $T$ be a bounded linear operator on a Banach space and $f$ an analytic function, defined on the spectrum of $T$. We study the relations between the rate of growth of the resolvent of $T$ and that of $f(T)$. We also discuss whether the property of unconditional basisness of eigenspaces or root spaces of $f(T)$ implies the corresponding property for $T$, and related issues.
We investigate matrices satisfying the Kreiss condition $$\|(zI-T)^{-1}\|\le\cfrac{K}{|z|-1}, \hspace{0.7 cm} |z|>1, $$ with $K$ lying arbitrarily close to $1.$ We provide lower bounds for the power growth of such matrices, which complement and refine related estimates due to Nikolski and Spijker-Tracogna-Welfert. We also study operators that satisfy a variant of the above Kreiss condition where $K$ is replaced by $1+ε(|z|)$, where the positive continuous function $ε(|z|)$ tends to $0$ as $|z|\to 1^+.$ We show that, if the spectrum of $T$ touches the unit circle only at a single point and the resolvent of $T$ satisfies a growth restriction along the unit circle, it is possible to choose $ε$ so that this Kreiss-type condition guarantees similarity to a contraction. At the core of our proof lies a positivity argument involving the double-layer potential operator. Counterexamples related to less restrictive choices of $ε$ are also provided.
Given a bounded open subset $Ω$ and closed subsets $A,B$ of $\mathbb{R}^k$, we discuss when an estimate $u(x)\le g(dist(x,A\cup B))$, $x\inΩ\setminus(A\cup B)$, for a function $u$ subharmonic on $Ω\setminus B$, implies that $u(x)\le h(dist(x,B))$, $x\inΩ\setminus B$, where $g,h:(0,\infty)\to (0,\infty)$ are decreasing functions and $g(0^+)=h(0^+)=\infty$. We seek for explicit expressions of $h$ in terms of $g$. We give some results of this type and show that Domar's work (On the existence of a largest subharmonic minorant of a given function, Ark. Mat., 3 (1957), pp. 429-440) permits one to deduce other results in this direction. Then we compare these two approaches. Similar results are deduced for estimates of analytic functions.
Using the model theory for Toeplitz operators with smooth symbols developed by the fourth author in the 80's, we study whether such operators $T_{F}$ can be embedded into a $C_{0}$-semigroup of operators on the Hardy space $H^p$ of the open unit disk, $1<p<\infty$. We show that it is the case as soon as $0$ belongs to the unbounded connected component of $\mathbb{C}$ minus the interior of the spectrum of $T_{F}$. We provide several conditions on the symbol $F$, both geometric and analytic in nature, ensuring that this sufficient condition is also necessary. For a certain class of symbols, where the curve $F(\mathbb{T})$ is a ``figure eight in a loop" such that $\mathbb{C}\setminusσ(T_F)$ has a bounded connected component, we obtain a complete characterization of the embeddability of $T_F$ into a $C_0$-semigroup. In the last part of the paper, we discuss the embeddability of $T_F$ when the symbol $F$ is not necessarily smooth, using connections with the numerical range and the functional calculus for bounded sectorial operators.
We provide a complete characterization of those non-elliptic semigroups of holomorphic self-maps of the unit disc for which the linear span of eigenvectors of the generator of the corresponding semigroup of composition operators is weak-star dense in $H^\infty$. We also give some necessary and some sufficient conditions for completeness in $H^p$. This problem is equivalent to the completeness of the corresponding exponential functions in $H^\infty$ (in the weak-star sense) or in $H^p$ of the Koenigs domain of the semigroup. As a tool needed for the results, we introduce and study discontinuities of semigroups of holomorphic self-maps of the unit disc.
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 Hardy spaces. Our result generalizes previous results in this context and it is related to cocycles of flows of analytic functions on Banach spaces. Likewise, for a large class of non-separable Banach spaces $X $ of analytic functions in $\mathbb{D}$ contained in the Bloch space, we prove that no non-trivial holomorphic flow induces a $C_0$-semigroup of weighted composition operators on $X$. This generalizes previous results regarding $C_0$-semigroup of (unweighted) composition operators.
For an invertible linear operator $T$ on a Hilbert space $H$, put \[ α(T^*,T) := -T^{*2}T^2 + (1+r^2) T^* T - r^2 I, \] 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 $α(t)=(1-t) (t-r^2)$. We give a concrete unitarily equivalent functional model for operators satisfying $α(T^*,T)\ge0$. In particular, we prove that the closed annulus $r\le |z|\le 1$ is a complete $K$-spectral set for $T$. We explain the relation of the model with the Sz.-Nagy--Foias one and with the observability gramian and discuss the relationship of this class with other operator classes related to the annulus.
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 encode in a closed-form the spectral behavior of $T$. Under mild geometric conditions on the spectral measure of $N$ and some smoothness constraints on $K$ we show that the operator $T$ admits invariant subspaces, or even it is decomposable.
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 algebras techniques and is an improvement of a recent result by the last two authors. We also prove that any $a$-contraction $T$ is a $b$-contraction, if $b<a$ and one imposes an additional condition on the growth of the norms of $T^n x$, where $x$ is an arbitrary vector. Here we use some properties of fractional finite differences.
We discuss when an operator, 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. To the contrary to the previous work, the kernel need not be of Nevanlinna-Pick type. We derive some consequences concerning the ergodic behavior of the operator.
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 on dilation to the boundary of the spectrum, along with the Dynkin functional calculus for smooth functions, which is based on pseudoanalytic continuation.
Let $T$ be a bounded linear operator on a Hilbert space $H$ such that \[ α[T^*,T]:=\sum_{n=0}^\infty α_n T^{*n}T^n\ge 0. \] where $α(t)=\sum_{n=0}^\infty α_n t^n$ is a suitable analytic function in the unit disc $\mathbb{D}$ with real coefficients. We prove that if $α(t) = (1-t) \tildeα (t)$, where $\tildeα$ has no roots in $[0,1]$, then $T$ is similar to a contraction. Operators of this type have been investigated by Agler, Müller, Olofsson, Pott and others, however, we treat cases where their techniques do not apply. We write down an explicit Nagy-Foias type model of an operator in this class and discuss its usual consequences (completeness of eigenfunctions, similarity to a normal operator, etc.). We also show that the limits of $\|T^nh\|$ as $n\to\infty$, $h\in H$, do not exist in general, but do exist if an additional assumption on $α$ is imposed. Our approach is based on a factorization lemma for certain weighted $\ell^1$ Banach algebras.
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 operators, so that the condition of quasicontractivity of the semigroup in the cited result is not necessary. Our result applies to a rather general class of Banach spaces of analytic functions in the unit disc.
Let $Φ$ be a family of functions analytic in some neighborhood of a complex domain $Ω$, and let $T$ be a Hilbert space operator whose spectrum is contained in $\overlineΩ$. Our typical result shows that under some extra conditions, if the closed unit disc is complete $K'$-spectral for $ϕ(T)$ for every $ϕ\in Φ$, then $\overlineΩ$ is complete $K$-spectral for $T$ for some constant $K$. In particular, we prove that under a geometric transversality condition, the intersection of finitely many $K'$-spectral sets for $T$ is again $K$-spectral for some $K\ge K'$. These theorems generalize and complement results by Mascioni, Stessin, Stampfli, Badea-Beckerman-Crouzeix and others. We also extend to non-convex domains a result by Putinar and Sandberg on the existence of a skew dilation of $T$ to a normal operator with spectrum in $\partialΩ$. As a key tool, we use the results from our previous paper on traces of analytic uniform algebras.
Given a complex domain $Ω$ and analytic functions $φ_1,\ldots,φ_n : Ω\to \mathbb{D}$, we give geometric conditions for $H^\infty(Ω)$ to be generated by functions of the form $g \circ φ_k$, $g \in H^\infty(\mathbb{D})$. We apply these results to the extension of bounded functions on an analytic one-dimensional complex subvariety of the polydisk $\mathbb{D}^n$ to functions in the Schur-Agler algebra of $\mathbb{D}^n$, with an estimate on the norm of the extension. Our proofs use some extension of the techniques of separation of singularities by Havin, Nersessian and Ortega-Cerdá.
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 matrix $A$. Given an operator $A$, for a class of cases, we find a matrix $B$ that provides an almost optimal decay rate. We show how our results can be applied to the problem of optimizing the decay rate for a large finite collection of control systems $(A, B_j)$, $j=1, \dots, N$, and illustrate this on an example of a concrete mechanical system. At the end of the article, we pose several questions concerning the decay rates in the context of linear quadratic optimization and in a more general context of the pole placement problem.
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 $L^2(-π, π)$. Analogous results are obtained for the systems of reproducing kernels in the de Branges spaces of entire functions. We also apply these results (via a functional model) to the spectral theory of rank one perturbations of compact self-adjoint operators.
We prove that a sectorial operator admits an H-infty - functional calculus if and only if it has a functional model of Nagy-Foias 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 McIntosh square function norm). We also show that this quadratic norm is close to the original one, in the sense that there is only a logarithmic gap between them.