Let A and M be closed linear operators defined on a complex Banach space X. Using operator-valued Fourier multipliers theorems, we obtain necessary and sufficient conditions to guarantee existence and uniqueness of periodic solutions to the equation d dt (Mu(t)) = Au(t) + f(t), in terms of either boundedness or R-boundedness of the modified resolvent operator determined by the equation. Our res...

Let T : C(S) → C(S) be a bounded linear operator. We present a necessary and sufficient condition for the so-called Daugavet equation ‖Id+ T ‖ = 1 + ‖T ‖ to hold, and we apply it to weakly compact operators and to operators factoring through c0. Thus we obtain very simple proofs of results by Foiaş, Singer, Pe lczyński, Holub and others. If E is a real Banach space, let us say that an operator ...

In ‎the present ‎paper, ‎we ‎introduce the ‎two-wavelet ‎localization ‎operator ‎for ‎the square ‎integrable ‎representation ‎of a‎ ‎homogeneous space‎ with respect to a relatively invariant measure. ‎We show that it is a bounded linear operator. We investigate ‎some ‎properties ‎of the ‎two-wavelet ‎localization ‎operator ‎and ‎show ‎that ‎it ‎is a‎ ‎compact ‎operator ‎and is ‎contained ‎in‎ a...

A spectral characterization of solutions of abstract linear differential equation systems is given. The characterization is in terms of the spectrum of a related continuous self-adjoint linear operator.

The problem of global illumination is virtually synonymouswith solving the rendering equation. Although a great deal of research has been directed toward Monte Carlo and finite element methods for solving the rendering equation, little is known about the continuous equation beyond the existence and uniqueness of its solution. The continuous problem may be posed in terms of linear operators acti...

The objective of the paper is to investigate the approximate controllability property of a linear stochastic control system with values in a separable real Hilbert space. In a first step we prove the existence and uniqueness for the solution of the dual linear backward stochastic differential equation. This equation has the particularity that in addition to an unbounded operator acting on the Y...

In this paper, we give the spectral theory for eigenvalues and eigenfunctions of a boundary value problem consisting of the linear fractional Bessel operator. Moreover, we show that this operator is self-adjoint, the eigenvalues of the problem are real, and the corresponding eigenfunctions are orthogonal. In this paper, we give the spectral theory for eigenvalues and eigenfunctions...

We study the nonexistence of global weak solutions to following semi-linear Moore - Gibson-Thompson equation with nonlinearity derivative type, namely,$$\left\{\begin{array}{l}u_{ttt}+u_{tt}-\Delta u-(-\Delta )^{\frac{\alpha}{2}}u_{t}=|u_t|^p,\quad x\in \R^n,\quad t>0,\\u(0,x)= u_0(x),\quad u_t(0,x)=u_1(x), \quad u_{tt}(0,x)= u_2(x) \R^n,\end{array}\right.$$where $\alpha\in (0, 2],\quad ...