Invertibility of linear difference operators with constant coefficients
Let \(X\) and \(Y\) be complex Banach spaces and \(\text{Hom}(X,Y)\) be the Banach space of all bounded linear operators \(A:X\to Y\). Given a locally compact Abelian group \(G\) with the group operation written additively, let \(\mathcal{F}(G,X)\) be one of the following spaces: \(L_p(G,X)\) \((1\leq p\leq\infty)\), \(C(G,X)\), or \(C_0(G,X)\). The author studies the invertibility of linear difference operators \(\mathcal{D}\in\text{Hom}(\mathcal{F}(G,X),\mathcal{F}(G,Y))\) and \(\mathcal{D}_+, \mathcal{D}_+^0\in\text{Hom}(\mathcal{F}(\mathbb{Z}_+,X),\mathcal{F}(\mathbb{Z}_+,Y))\) given by \[ (\mathcal{D}x)(g)=Ax(g)-Bx(g-g_0), \quad g,g_0\in G, \quad x\in\mathcal{F}(G,X), \] \[ (\mathcal{D}_+x)(k)=Ax(k)-Bx(k+1), \quad k\in\mathbb{Z}_+,\quad x\in\mathcal{F}(\mathbb{Z}_+,X), \] \[ (\mathcal{D}_+^0x)(k)=\begin{cases} Ax(0), & k=0,\\ Ax(k)-Bx(k-1), & k\geq 1, \end{cases} x\in \mathcal{F}(\mathbb{Z}_+,X), \] where \(A,B\in \text{Hom}(X,Y)\) and \(\mathbb{Z}_+=\{0,1,2,\ldots\}\). Such operators arise in the theory of differential-difference equations. For all these operators, invertibility criteria and formulas for the inverse operators are established. Sufficient conditions for the right invertibility of \(\mathcal{D}_+\) and for the left invertibility of \(\mathcal{D}_+^0\) as well as formulas for their one-sided inverses are obtained. The kernel of \(\mathcal{D}_+\) and the image of \(\mathcal{D}_+^0\) are described.
Hankel-Schur and Toeplitz-Schur multipliers
scientific article; zbMATH DE number 1752656