Pedro José Miana Sanz

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 $\ell^p$.

We show that the composition hyperbolic group in the unit disc, once transferred to act on sequence spaces, is bounded on $$\ell^p$$ ℓ p if and only if $${p=2}$$ p = 2 . We introduce some integral operators subordinated to that group which are natural generalizations of classical operators on sequences. For the description of such operators, we use some combinatorial identities which look interesting in their own.

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.

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.

In this paper, we study the solution of the quadratic equation $$TY^2-Y+I=0$$TY2-Y+I=0 whereTis a linear and bounded operator on a Banach spaceX. We describe the spectrum set and the resolvent operator ofYin terms of the ones ofT. In the case that 4Tis a power-bounded operator, we show that a solution (named Catalan generating function) of the above equation is given by the Taylor series $$\begin{aligned} C(T):=\sum _{n=0}^\infty C_nT^n, \end{aligned}$$C(T):=∑n=0∞CnTn, where the sequence $$(C_n)_{n\ge 0}$$(Cn)n≥0 is the well-known Catalan numbers sequence. We expressC(T) by means of an integral representation which involves the resolvent operator $$(\lambda T)^{-1}$$(λT)-1 . Some particular examples to illustrate our results are given, in particular an iterative method defined for square matricesTwhich involves Catalan numbers.

In this paper, we treat some fractional differential equations on the sequence Lebesgue spaces ${\ell }^p\left({\mathbb{N}}_0\right)$ with $p\ge 1$. The Caputo fractional calculus extends the usual derivation. The operator, associated to the Cauchy problem, is defined by a convolution with a sequence of compact support and belongs to the Banach algebra ${\ell }^1\left(\mathbb{Z}\right)$. We treat in detail some of these compact support sequences. We use techniques from Banach algebras and a Functional Analysis to explicity check the solution of the problem.

In this paper we study solutions of the quadratic equation $$AY^2-Y+I=0$$AY2-Y+I=0 whereAis the generator of a one parameter family of operator ( $$C_0$$C0 -semigroup or cosine functions) on a Banach spaceXwith growth bound $$w_0 \le \frac{1}{4}$$w0≤14 . In the case of $$C_0$$C0 -semigroups, we show that a solution, which we call Catalan generating function ofA,C(A), is given by the following Bochner integral, $$\begin{aligned} C(A)x := \int _{0}^\infty c(t) T(t)x \; \mathrm{d}t, \quad x\in X, \end{aligned}$$C(A)x:=∫0∞c(t)T(t)xdt,x∈X, wherecis the Catalan kernel, $$\begin{aligned} c(t) := \frac{1}{2\pi } \int _{\frac{1}{4}}^\infty e^{-\lambda t} \frac{\sqrt{4\lambda -1}}{\lambda } \; \mathrm{d}\lambda , \quad t>0. \end{aligned}$$c(t):=12π∫14∞e-λt4λ-1λdλ,t>0. Similar (and more complicated) results hold for cosine functions. We study algebraic properties of the Catalan kernelcas an element in Banach algebras $$L^1_{\omega }(\mathbb R^+)$$Lω1(R+) , endowed with the usual convolution product, $$*$$∗ and with the cosine convolution product, $$*_c$$∗c . The Hille–Phillips functional calculus allows to transfer these properties to $$C_0$$C0 -semigroups and cosine functions. In particular, we obtain a spectral mapping theorem forC(A). Finally, we present some examples, applications and conjectures to illustrate our results.