arXiv subjects

Daniel Estévez

Publications and source records attributed to Daniel Estévez.

5 recordsLinked to original sources

arXiv2018

Resolvent criteria for similarity to a normal operator with spectrum on a curve

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.

math.FA↗

arXiv2017

Tests for complete $K$-spectral sets

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.

math.FA↗

arXiv2017

Traces of analytic uniform algebras on subvarieties and test collections

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.

math.CV↗

arXiv2014

Explicit traces of functions on Sobolev spaces and quasi-optimal linear interpolators

Let $Λ\subset R$ be a strictly increasing sequence. For $r = 1,2$, we give a simple explicit expression for an equivalent norm on the trace spaces $W_p^r(R)|_Λ$, $L_p^r(R)|_Λ$ of the non-homogeneous and homogeneous Sobolev spaces with $r$ derivatives $W_p^r(R)$, $L_p^r(R)$. We also construct an interpolating spline of low degree having optimal norm up to a constant factor. A general result relating interpolation in $L^r_p(R)$ and $W^r_p(R)$ for all $r \geq 1$ is also given.

math.FA↗

arXiv2013

Decay rate estimations for linear quadratic optimal regulators

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.

math.OC↗