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Yuri Tomilov

Publications and source records attributed to Yuri Tomilov.

At least 37 records · Page 2Linked to original sources
arXiv2018

Joint numerical ranges and compressions of powers of operators

We identify subsets of the joint numerical range of an operator tuple in terms of its joint spectrum. This result helps us to transfer weak convergence of operator orbits into certain approximation and interpolation properties for powers in the uniform operator topology. This is a far-reaching generalization of one of the main results in our recent paper posted as arXiv:1607.00040. Moreover, it yields an essential (but partial) generalization of Bourin's "pinching" theorem. It also allows us to revisit several basic results on joint numerical ranges, provide them with new proofs and find a number of new results.

math.FA↗
arXiv2017

Circles in the spectrum and the geometry of orbits: a numerical ranges approach

We prove that a bounded linear Hilbert space operator has the unit circle in its essential approximate point spectrum if and only if it admits an orbit satisfying certain orthogonality and almost-orthogonality relations. This result is obtained via the study of numerical ranges of operator tuples where several new results are also obtained. As consequences of our numerical ranges approach, we derive in particular wide generalizations of Arveson's theorem as well as show that the weak convergence of operator powers implies the uniform convergence of their compressions on an infinite-dimensional subspace. Several related results have been proved as well.

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arXiv2016

Resolvent representations for functions of sectorial operators

We obtain integral representations for the resolvent of $ψ(A)$, where $ψ$ is a holomorphic function mapping the right half-plane and the right half-axis into themselves, and $A$ is a sectorial operator on a Banach space. As a corollary, for a wide class of functions $ψ$, we show that the operator $-ψ(A)$ generates a sectorially bounded holomorphic $C_0$-semigroup on a Banach space whenever $-A$ does, and the sectorial angle of $A$ is preserved. When $ψ$ is a Bernstein function, this was recently proved by Gomilko and Tomilov, but the proof here is more direct. Moreover, we prove that such a permanence property for $A$ can be described, at least on Hilbert spaces, in terms of the existence of a bounded $H^{\infty}$-calculus for $A$. As byproducts of our approach, we also obtain new results on functions mapping generators of bounded semigroups into generators of holomorphic semigroups and on subordination for Ritt operators.

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arXiv2016

On discrete subordination of power bounded and Ritt operators

By means of a new technique, we develop further a discrete subordination approach to the functional calculus of power bounded and Ritt operators initiated by N. Dungey in [19]. This allows us to show, in particular, that (infinite) convex combinations of powers of Ritt operators are Ritt. Moreover, we provide a unified framework for several main results on discrete subordination from [19] and answer a question left open in [19]. The paper can be considered as a complement to [26] for the discrete setting.

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arXiv2016

Fine scales of decay of operator semigroups

Motivated by potential applications to partial differential equations, we develop a theory of fine scales of decay rates for operator semigroups. The theory contains, unifies, and extends several notable results in the literature on decay of operator semigroups and yields a number of new ones. Its core is a new operator-theoretical method of deriving rates of decay combining ingredients from functional calculus, and complex, real and harmonic analysis. It also leads to several results of independent interest.

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arXiv2016

L^p-tauberian theorems and L^p-rates for energy decay

We prove $L^p$-analogues of the classical tauberian theorem of Ingham and Karamata, and its variations giving rates of decay. These results are applied to derive $L^p$-decay of operator families arising in the study of the decay of energy for damped wave equations and local energy for wave equations in exterior domains. By constructing some examples of critical behaviour we show that the $L^p$-rates of decay obtained in this way are best possible under our assumptions.

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arXiv2015

The index formula and the spectral shift function for relatively trace class perturbations

We compute the Fredholm index, ${\rm ind}(D_A)$, of the operator $D_A = (d/dt) + A$ on $L^2(\mathbb{R};\mathcal{H})$ associated with the operator path $\{A(t)\}_{t=-\infty}^{\infty}$, where $(A f)(t) = A(t) f(t)$ for a.e. $t\in\mathbb{R}$, and appropriate $f \in L^2(\mathbb{R};\mathcal{H})$, via the spectral shift function $ξ(\, \cdot \,;A_+,A_-)$ associated with the pair $(A_+, A_-)$ of asymptotic operators $A_{\pm}=A(\pm\infty)$ on the separable complex Hilbert space $\mathcal{H}$ in the case when $A(t)$ is generally an unbounded (relatively trace class) perturbation of the unbounded self-adjoint operator $A_-$. We derive a formula (an extension of a formula due to Pushnitski) relating the spectral shift function $ξ(\, \cdot \,;A_+,A_-)$ for the pair $(A_+, A_-)$, and the corresponding spectral shift function $ξ(\, \cdot \,;H_2,H_1)$ for the pair of operators $(H_2,H_1)=(D_A {D_A}^*, {D_A}^* D_A)$ in this relative trace class context. This formula is then used to identify the Fredholm index of $D_A$ with $ξ(0;A_+,A_-)$. In addition, we prove that ${\rm ind}(D_A)$ coincides with the spectral flow SpFlow$(\{A(t)\}_{t=-\infty}^\infty)$ of the family $\{A(t)\}_{t\in\mathbb{R}}$ and also relate it to the (Fredholm) perturbation determinant for the pair $(A_+, A_-)$. We also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta-invariant.

math.SP↗
arXiv2014

Generation of subordinated holomorphic semigroups via Yosida's theorem

Using functional calculi theory, we obtain several estimates for $\|ψ(A)g(A)\|$, where $ψ$ is a Bernstein function, $g$ is a bounded completely monotone function and $-A$ is the generator of a holomorphic $C_0$-semigroup on a Banach space, bounded on $[0,\infty)$. Such estimates are of value, in particular, in approximation theory of operator semigroups. As a corollary, we obtain a new proof of the fact that $-ψ(A)$ generates a holomorphic semigroup whenever $-A$ does, established recently in [8] by a different approach.

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arXiv2014

Product formulas in functional calculi for sectorial operators

We study the product formula $(fg)(A) = f(A)g(A)$ in the framework of (unbounded) functional calculus of sectorial operators $A$. We give an abstract result, and, as corollaries, we obtain new product formulas for the holomorphic functional calculus, an extended Stieltjes functional calculus and an extended Hille-Phillips functional calculus. Our results generalise previous work of Hirsch, Martinez and Sanz, and Schilling.

math.FA↗
arXiv2014

On the Witten index in terms of spectral shift functions

We study the model operator $\mathbf{D}_{\mathbf{A}} = (d/dt) + \mathbf{A}$ in $L^2(\mathbb{R};\mathcal{H})$ associated with the operator path $\{A(t)\}_{t=-\infty}^{\infty}$, where $(\mathbf{A} f)(t) = A(t) f(t)$ for a.e.\ $t\in\mathbb{R}$, and appropriate $f \in L^2(\mathbb{R};\mathcal{H})$ (with $\mathcal{H}$ a separable, complex Hilbert space). Denoting by $A_{\pm}$ the norm resolvent limits of $A(t)$ as $t \to \pm \infty$, our setup permits $A(t)$ in $\mathcal{H}$ to be an unbounded, relatively trace class perturbation of the unbounded self-adjoint operator $A_-$, and no discrete spectrum assumptions are made on $A_{\pm}$. We introduce resolvent and semigroup regularized Witten indices of $\mathbf{D}_{\mathbf{A}}$, denoted by $W_r$ and $W_s$, and prove that these regularized indices coincide with the Fredholm index of $\mathbf{D}_{\mathbf{A}}$ whenever the latter is Fredholm. In situations where $\mathbf{D}_{\mathbf{A}}$ ceases to be a Fredholm operator in $L^2(\mathbb{R};\mathcal{H})$ we compute its resolvent (resp., semigroup) regularized Witten index in terms of the spectral shift function $ξ(\,\cdot\,;A_+,A_-)$ associated with the pair $(A_+, A_-)$ as follows: Assuming $0$ to be a right and a left Lebesgue point of $ξ(\,\cdot\,\, ; A_+, A_-)$, denoted by $ξ_L(0_+; A_+,A_-)$ and $ξ_L(0_-; A_+, A_-)$, we prove that $0$ is also a right Lebesgue point of $ξ(\,\cdot\,\, ; \mathbf{H_2}, \mathbf{H_1})$, denoted by $ξ_L(0_+; \mathbf{H_2}, \mathbf{H_1})$, and that \begin{align*} W_r(\mathbf{D}_{\mathbf{A}}) &= W_s(\mathbf{D}_{\mathbf{A}}) \\ & = ξ_L(0_+; \mathbf{H_2}, \mathbf{H_1}) \\ & = [ξ_L(0_+; A_+,A_-) + ξ_L(0_-; A_+, A_-)]/2, \end{align*} the principal result of this paper. In the special case where $\dim(\mathcal{H}) < \infty$, we prove that the Witten indices of $\mathbf{D}_{\mathbf{A}}$ are either integer, or half-integer-valued.

math.SP↗
arXiv2014

On subordination of holomorphic semigroups

We prove that for any Bernstein function $ψ$ the operator $-ψ(A)$ generates a holomorphic $C_0$-semigroup $(e^{-tψ(A)})_{t \ge 0}$ on a Banach space, whenever $-A$ does. This answers a question posed by Kishimoto and Robinson. Moreover, giving a positive answer to a question by Berg, Boyadzhiev and de Laubenfels, we show that $(e^{-tψ(A)})_{t \ge 0}$ is holomorphic in the holomorphy sector of $(e^{-tA})_{t \ge 0},$ and if $(e^{-tA})_{t \ge 0}$ is sectorially bounded in this sector then $(e^{-tψ(A)})_{t \ge 0}$ has the same property. We also obtain new sufficient conditions on $ψ$ in order that, for every Banach space $X$, the semigroup $(e^{-tψ(A)})_{t\ge 0}$ on $X$ is holomorphic whenever $(e^{-tA})_{t\ge 0}$ is a bounded $C_0$-semigroup on $X$. These conditions improve and generalize well-known results by Carasso-Kato and Fujita.

math.FA↗
arXiv2014

Some Operator Bounds Employing Complex Interpolation Revisited

We revisit and extend known bounds on operator-valued functions of the type $$ T_1^{-z} S T_2^{-1+z}, \quad z \in \ol Σ= \{z\in\bbC\,|\, \Re(z) \in [0,1]\}, $$ under various hypotheses on the linear operators $S$ and $T_j$, $j=1,2$. We particularly single out the case of self-adjoint and sectorial operators $T_j$ in some separable complex Hilbert space $\cH_j$, $j=1,2$, and suppose that $S$ (resp., $S^*$) is a densely defined closed operator mapping $\dom(S) \subseteq \cH_1$ into $\cH_2$ (resp., $\dom(S^*) \subseteq \cH_2$ into $\cH_1$), relatively bounded with respect to $T_1$ (resp., $T_2^*$). Using complex interpolation methods, a generalized polar decomposition for $S$, and Heinz's inequality, the bounds we establish lead to inequalities of the following type, \begin{align*} & \big\|\ol{T_2^{-x}ST_1^{-1+x}}\big\|_{\cB(\cH_1,\cH_2)} \leq N_1 N_2 e^{(θ_1 + θ_2) [x(1-x)]^{1/2}} \\ & \quad \times \big\|ST_1^{-1}\big\|_{\cB(\cH_1,\cH_2)}^{1-x} \, \big\|S^*(T_2^*)^{-1}\big\|_{\cB(\cH_2,\cH_1)}^{x}, \quad x \in [0,1], \end{align*} assuming that $T_j$ have bounded imaginary powers, that is, for some $N_j\ge 1$ and $θ_j \ge 0,$ $$ \big\|T_j^{is}\big\|_{\cB(\cH)} \leq N_j e^{θ_j |s|}, \quad s \in \bbR, \; j=1,2. $$ We also derive analogous bounds with $\cB(\cH_1,\cH_2)$ replaced by trace ideals, $\cB_p(\cH_1, \cH_2)$, $p \in [1,\infty)$. The methods employed are elementary, predominantly relying on Hadamard's three-lines theorem and Heinz's inequality.

math.FA↗
arXiv2014

A Jost-Pais-type reduction of Fredholm determinants and some applications

We study the analog of semi-separable integral kernels in $\cH$ of the type {equation*} K(x,x')={cases} F_1(x)G_1(x'), & a<x'< x< b, \\ F_2(x)G_2(x'), & a<x<x'<b, {cases} {equation*} where $-\infty\leq a<b\leq \infty$, and for a.e.\ $x \in (a,b)$, $F_j (x) \in \cB_2(\cH_j,\cH)$ and $G_j(x) \in \cB_2(\cH,\cH_j)$ such that $F_j(\cdot)$ and $G_j(\cdot)$ are uniformly measurable, and {equation*} \|F_j(\cdot)\|_{\cB_2(\cH_j,\cH)} \in L^2((a,b)), \; \|G_j (\cdot)\|_{\cB_2(\cH,\cH_j)} \in L^2((a,b)), \quad j=1,2, {equation*} with $\cH$ and $\cH_j$, $j=1,2$, complex, separable Hilbert spaces. Assuming that $K(\cdot, \cdot)$ generates a trace class operator $\bsK$ in $L^2((a,b);\cH)$, we derive the analog of the Jost-Pais reduction theory that succeeds in proving that the Fredholm determinant ${\det}_{L^2((a,b);\cH)}(\bsI - α\bsK)$, $α\in \bbC$, naturally reduces to appropriate Fredholm determinants in the Hilbert spaces $\cH$ (and $\cH_1 \oplus \cH_2$). Explicit applications of this reduction theory are made to Schrödinger operators with suitable bounded operator-valued potentials. In addition, we provide an alternative approach to a fundamental trace formula first established by Pushnitski which leads to a Fredholm index computation of a certain model operator.

math.FA↗
arXiv2013

On convergence rates in approximation theory for operator semigroups

We create a new, functional calculus, approach to approximation of C_0-semigroups on Banach spaces. As an application of this approach, we obtain optimal convergence rates in classical approximation formulas for C_0-semigroups. In fact, our methods allow one to derive a number of similar formulas and equip them with sharp convergence rates. As a byproduct, we prove a new interpolation principle leading to efficient norm estimates in the Banach algebra of Laplace transforms of bounded measures on the semi-axis.

math.FA↗
arXiv2013

On rates in Euler's formula for C_0-semigroups

By functional calculus methods, we obtain optimal convergence rates in Euler's approximation formula for C_0-semigroups restricted to ranges of generalized Stieltjes functions. Our results include a number of partial cases studied in the literature and cannot essentially be improved.

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arXiv2011

Bernstein functions and rates in mean ergodic theorems for operator semigroups

We present a functional calculus approach to the study of rates of decay in mean ergodic theorems for bounded strongly continuous operator semigroups. A central role is played by operators of the form $g(A)$, where $-A$ is the generator of the semigroup and $g$ is a Bernstein function. In addition, we obtain some new results on Bernstein functions that are of independent interest.

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