arXiv subjects

Pedro J. Miana

Publications and source records attributed to Pedro J. Miana.

At least 19 recordsLinked to original sources

arXiv2026

Entire groups generated by fractional powers of operators

Let $T$ be a power-bounded operator on a Banach space $X$. We treat the sequence of polynomials $(p_{n;a})_{n\ge 0}$ such that the entire group generated by the fractional power operator $-(I-T)^a$ is given by $$ e^{-t(I-T)^a}=e^{-t}\sum_{n=0}^\infty p_{n;a}(t)T^n, \qquad t\in \CC, \quad \Re a>0. $$ We provide a self-contained introduction to the polynomial family $(p_{n;a})_{n\ge 0}$, for $a\in \CC$, whose coefficients are determined by means of a suitable recurrence relation. The sequence $(n!p_{n;a})_{n\ge 0}$ forms a family of Sheffer polynomials. For $\Re a>0$ and $t\in \CC$, the sequence $(p_{n;a}(t))_{n\ge 0}$ belongs to the Lebesgue sequence space $\ell^1$ of absolutely summable sequences. Moreover, these polynomials are closely related to the Lévy density functions $(f_{t,α})_{t>0}$ defined for $0<α<1$. Finally, we discuss several particular cases corresponding to specific values of $a\in \CC$, as well as applications to fractional powers in the Banach algebra $\ell^1$, multiplication operators, and Cesàro means.

math.FA↗

arXiv2026

Cesàro-Hardy operators on $L^p[0,1]$: fine spectrum, weighted Koopman semigroups and invariant subspaces

In this paper we study boundedness and detailed spectral properties for the Cesàro-Hardy operator and some generalizations in $L^p[0,1]$. The study employs $C_0$-semigroup theory, expressing the Cesàro-Hardy operators and their dual operators through subordination with $C_0$-semigroups $T(t)$ and $S(t)$ respectively. The spectral properties of the semigroup's infinitesimal generators are transferred to the Cesàro-Hardy operators using functional calculus methods. Furthermore, some implications for the Invariant Subspace Problem are explored by demonstrating the universality of certain translations related to the semigroup $T(t)$, and providing results on the invariant subspaces of these operators.

math.FA↗

arXiv2025

Two Koopman semigroups on discrete Lebesgue spaces

In this paper we are interested to connect Koopman semigroups in Lebesgue funcion spaces $L^p(\R^+)$ and $C_0$-semigroups in Lebesgue sequence spaces $\ell^p$ for $1\le p < \infty$. To get this we use certain Poisson transformation $¶: L^p(\mathbb{R}^+)\to \ell^p$ and its adjoint $¶^*$ which allows carry semigroup properties from one space to the other one. Two Koopman semigroups on $\ell^p$ are presented and linked to the standard Koopman semigroup $T_p(t)f(r):= e^{-{t\over p}}f(e^{-t}r)$ and $S_{p}(t)f(r):= e^{-t\over p}f(e^{-t}r+1-e^{-t})$ for $t,r>0$ on $L^p(\R^+)$. In the last section we introduce Cesàro-like operators subordinated to these Koopman semigroups on $\ell^p$.

math.FA↗

arXiv2024

RKH spaces of Brownian type defined by Cesàro-Hardy operators

We study reproducing kernel Hilbert spaces introduced as ranges of generalized Cesàro-Hardy operators, in one real variable and in one complex variable. Such spaces can be seen as formed by absolutely continuous functions on the positive half-line (or paths of infinite length) of fractional order, in the real case. A theorem of Paley-Wiener type is given which connects the real setting with the complex one. These spaces are related with fractional operations in the context of integrated Brownian processes. We give estimates of the norms of the corresponding reproducing kernels.

math.FA↗

arXiv2024

Hilbertian Hardy-Sobolev spaces on a half-plane

In this paper we deal with a scale of reproducing kernel Hilbert spaces $H^{(n)}_2$, $n\ge 0$, which are linear subspaces of the classical Hilbertian Hardy space on the right-hand half-plane $\mathbb{C}^+$. They are obtained as ranges of the Laplace transform in extended versions of the Paley-Wiener theorem which involve absolutely continuous functions of higher degree. An explicit integral formula is given for the reproducing kernel $K_{z,n}$ of $H^{(n)}_2$, from which we can find the estimate $\Vert K_{z,n}\Vert\sim\vert z\vert^{-1/2}$ for $z\in\mathbb{C}^+$. Then composition operators $C_φ:H_2^{(n)} \to H_2^{(n)}$, $C_φf=f\circ φ$, on these spaces are discussed, giving some necessary and some sufficient conditions for analytic maps $φ: \mathbb{C}^+\to \mathbb{C}^+$ to induce bounded composition operators.

math.FA↗

arXiv2024

Large time behaviour for the heat equation on $\Z,$ moments and decay rates

The paper is devoted to understand the large time behaviour and decay of the solution of the discrete heat equation in the one dimensional mesh $\Z$ on $\ell^p$ spaces, and its analogies with the continuous-space case. We do a deep study of the moments of the discrete gaussian kernel (which is given in terms of Bessel functions), in particular the mass conservation principle; that is reflected on the large time behaviour of solutions. We prove asymptotic pointwise and $\ell^p$ decay results for the fundamental solution. We use that estimates to get rates on the $\ell^p$ decay and large time behaviour of solutions. For the $\ell^2$ case, we get optimal decay by use of Fourier techniques.

math.AP↗

arXiv2024

Catalan generating functions for bounded operators

In this paper we study the solution of the quadratic equation $TY^2-Y+I=0$ where $T$ is a linear and bounded operator on a Banach space $X$. We describe the spectrum set and the resolvent operator of $Y$ in terms of operator $T$. In the case that $ 4T$ is a power-bounded operator, we show that a solution (named Catalan generating function) is given by the Taylor series $$ C(T):=\sum_{n=0}^\infty C_nT^n, $$ where the sequence $(C_n)_{n\ge 0}$ is the well-known Catalan numbers. We express $C(T)$ by means of an integral representation which involves the resolvent operator $(λ-T)^{-1}$. Some particular examples to illustrate our results are given, in particular an iterative method defined for square matrices $T$ which involves Catalan numbers.

math.FA↗

arXiv2024

Powers of Catalan generating functions for bounded operators

Let $c=(C_n)_{n\ge 0}$ be the Catalan sequence and $T$ a linear and bounded operator on a Banach space $X$ such $4T$ is a power-bounded operator. The Catalan generating function is defined by the following Taylor series, $$ C(T):=\sum_{n=0}^\infty C_nT^n. $$ Note that the operator $C(T)$ is a solution of the quadratic equation $TY^2-Y+I=0.$ In this paper we define powers of the Catalan generating function $C(T)$ in terms of the Catalan triangle numbers. We obtain new formulae which involve Catalan triangle numbers; the spectrum of $c^{\ast j}$ and the expression of $c^{-\ast j}$ for $j\ge 1$ in terms of Catalan polynomials ($\ast$ is the usual convolution product in sequences). In the last section, we give some particular examples to illustrate our results and some ideas to continue this research in the future.

math.FA↗

arXiv2019

Soñando con números, María Andresa Casamayor (1720-1780)

María Andresa Casamayor de la Coma, born in Zaragoza, is known as the first woman who published a scientific book in Spain. In this paper we provide answers to several of the most important questions about her unknown biography such as her birth day, the origins of her family, the houses where she and her close family lived, her job as a teacher and even her true name.

math.HO↗

arXiv2019

Generalized Stieltjes and other integral operators on Sobolev-Lebesgue spaces

For $μ>β>0$, the generalized Stieltjes operators $$ \mathcal{S}_{β,μ} f(t):={t^{μ-β}}\int_0^\infty {s^{β-1}\over (s+t)^μ}f(s)ds, \qquad t>0, $$ defined on Sobolev spaces $\mathcal{T}_p^{(α)}(t^α)$ (where $α\ge 0$ is the fractional order of derivation and these spaces are embedded in $L^p(\RR^+)$ for $p\ge 1$) are studied in detail. If $0 < β- \pp < μ$, then operators $\mathcal{S}_{β,μ}$ are bounded (and we compute their operator norms which depend on $p$); commute and factorize with generalized Cesáro operator on $\mathcal{T}_p^{(α)}(t^α)$ . We calculate and represent explicitly their spectrum set $σ(\mathcal{S}_{β,μ})$. The main technique is to subordinate these operators in terms of $C_0$-groups and transfer new properties from some special functions to Stieltjes operators. We also prove some similar results for generalized Stieltjes operators $ \mathcal{S}_{β,μ}$ in the Sobolev-Lebesgue $\mathcal{T}_p^{(α)}(\vert t\vert^α)$ defined on the real line $\R$. We show connections with the Fourier and the Hilbert transform and a convolution product defined by the Hilbert transform.

math.FA↗

arXiv2017

Zoel García de Galdeano y su presencia en sociedades matemáticas de su época

In this paper, we present the work of international mathematical associationism carried out by the mathematician from Navarra (Spain) Zoel García de Galdeano (1846-1924) during over 30 years. Garcia de Galdeano was a member of many of the most important mathematical associations of his time, thus being able to be in touch with foreign mathematicians of high level and to exchange ideas, works and collaborations. This work of associationism has been unnoticed in biographical and scientific studies up to now and helps to complete the picture of García de Galdeano as an international mathematician and as a wise importer of the main mathematical achievements of his time.

math.HO↗

arXiv2017

Generalized Cesàro operators, fractional finite differences and Gamma functions

In this paper, we present a complete spectral research of generalized Cesàro operators on Sobolev-Lebesgue sequence spaces. The main idea is to subordinate such operators to suitable $C_0$-semigroups on these sequence spaces. We introduce that family of sequence spaces using the fractional finite differences and we prove some structural properties similar to classical Lebesgue sequence spaces. In order to show the main results about fractional finite differences, we state equalities involving sums of quotients of Euler's Gamma functions. Finally, we display some graphical representations of the spectra of generalized Cesàro operators.

math.FA↗

arXiv2016

On well-posedness of vector-valued fractional differential-difference equations

We develop an operator-theoretical method for the analysis on well posedness of partial differential equations that can be modeled in the form \begin{equation*} \left\{ \begin{array}{rll} Δ^α u(n) &= Au(n+2) + f(n,u(n)), \quad n \in \mathbb{N}_0, \,\, 1< α\leq 2; u(0) &= u_0; u(1) &= u_1, \end{array} \right. \end{equation*} where $A$ is an closed linear operator defined on a Banach space $X$. Our ideas are inspired on the Poisson distribution as a tool to sampling fractional differential operators into fractional differences. Using our abstract approach, we are able to show existence and uniqueness of solutions for the problem (*) on a distinguished class of weighted Lebesgue spaces of sequences, under mild conditions on strongly continuous sequences of bounded operators generated by $A,$ and natural restrictions on the nonlinearity $f$. Finally we present some original examples to illustrate our results.

math.AP↗

arXiv2016

On stability of $N$-times integrated semigroups with nonquasianalytic growth

We discuss the behaviour at infinity of $n$-times integrated semigroups with nonquasianalytic growth and invertible generator. The results obtained extend in this setting a theorem of O. El Mennaoui on stability of bounded once integrated semigroups, and (partially) a theorem of Q. P. V$\tilde{\rm u}$ on stability of $C_0$-semigroups.

math.FA↗

arXiv2016

Sums of powers of Catalan triangle numbers

In this paper we consider combinatorial numbers $C_{m, k}$ for $m\ge 1$ and $k\ge 0$ which unifies the entries of the Catalan triangles $ B_{n, k}$ and $ A_{n, k}$ for appropriate values of parameters $m$ and $k$, i.e., $B_{n, k}=C_{2n,n-k}$ and $A_{n, k}=C_{2n+1,n+1-k}$. In fact, some of these numbers are the well-known Catalan numbers $C_n$ that is $C_{2n,n-1}=C_{2n+1,n}=C_n$. We present new identities for recurrence relations, linear sums and alternating sum of $C_{m,k}$. After that, we check sums (and alternating sums) of squares and cubes of $C_{m,k}$ and, consequently, for $ B_{n, k}$ and $ A_{n, k}$. In particular, one of these equalities solves an open problem posed in \cite{[GHMN]}. We also present some linear identities involving harmonic numbers $H_n$ and Catalan triangles numbers $C_{m,k}$. Finally, in the last section new open problems and identities involving $C_n$ are conjectured.

math.NT↗

arXiv2015

Sharp extensions and algebraic properties for solution families of vector-valued differential equations

In this paper we show the unexpected property that extension from local to global without loss of regularity holds for the solutions of a wide class of vector-valued differential equations, in particular for the class of fractional abstract Cauchy problems in the subdiffusive case. The main technique is the use of the algebraic structure of these solutions, which are defined by new versions of functional equations defining solution families of bounded operators. The convolution product and the double Laplace transform for functions of two variables are useful tools which we apply also to extend these solutions. Finally we illustrate our results with different concrete examples.

math.FA↗

arXiv2015

Cesáro sums and algebra homomorphisms of bounded operators

Let $X$ be a complex Banach space. The connection between algebra homomorphisms defined on subalgebras of the Banach algebra $\ell^{1}(\mathbb{N}_0)$ and the algebraic structure of Cesàro sums of a linear operator $T\in \mathcal{B}(X)$ is established. In particular, we show that every $(C, α)$-bounded operator $T$ induces - and is in fact characterized - by such an algebra homomorphism. Our method is based on some sequence kernels, Weyl fractional difference calculus and convolution Banach algebras that are introduced and deeply examined. To illustrate our results, improvements to bounds for Abel means, new insights on the $(C,α)$ boundedness of the resolvent operator for temperated $α$-times integrated semigroups, and examples of bounded homomorphisms are given in the last section.

math.FA↗

arXiv2014

A subordination principle on Wright functions and regularized resolvent families

We obtain a vector-valued subordination principle for $(g_α, g_α)$-regularized resolvent families which unified and improves various previous results in the literature. As a consequence we establish new relations between solutions of different fractional Cauchy problems. To do that, we consider scaled Wright functions which are related to Mittag-Leffler functions, the fractional calculus and stable Lévy processes. We study some interesting properties of these functions such as subordination (in the sense of Bochner), convolution properties, and their Laplace transforms. Finally we present some examples where we apply these results.

math.FA↗