Invertibility conditions and bounds for spectra of matrix integral operators (Q1969700)

Let \({\mathfrak H}\) be a Hilbert space with inner product \((.|.)_{\mathfrak H}\) and norm \(\|.\|_{\mathfrak H}\). If \(A\) is a linear operator on \({\mathfrak H}\), then the spectrum of \(A\) is denoted by \(\sigma(A)\), the eigenvalues are denoted by \(\lambda_k= \lambda_k(A)\), \(k= 1,2,3,\dots\)\ . As indicated in the introduction of this paper, the author derives estimates for the spectrum of a matrix integral operator by applying recently derived estimates for the norm of the resolvent. The results are particularly applicable to a continuous operator \(A\) in \(L^2({\mathfrak w},\mathbb{C}^n)\) defined by \(A(h)(x)=\widetilde Q(h)(x)+ \widetilde K(h)(x)\), \(\widetilde Q(h)(x)= Q(x)h(x)\), \[ \widetilde K(h)(x)= \int_{\mathfrak w} K(x,s) h(s) ds, \] where \(Q\), \(K\) are matrix-valued functions defined respectively on \({\mathfrak w}\), \({\mathfrak w}\times{\mathfrak w}\). A general form of two of the main theorems of the paper states that: if \[ g(A)= \Biggl(N^2(A)- \sum^k_{j= 1}|\lambda_j(A)|^2\Biggr)^{{1\over 2}}, \] where \(N^2(A)= \text{trace}(AA^*)\), \(g_0= \sup\{g(Q(x)),x\in{\mathfrak w}\}\), \[ \Gamma_0(z)= \sum^{n-1}_{k=1} k!^{-{1\over 2}} g^k_0 z^{-k-1}, \] \(F(z)\) has the form \(2^{{1\over 2}}\|\widetilde T\|_{\mathfrak H}\Gamma_0(z) \exp(\Gamma^2_0(z) N^2(V_{up}))\) and \(z(F)\) is the unique positive extreme right root of the equation \(F(z)= 1\), then for any \(\mu\) in \(\sigma(A)\), there is \(x\in{\mathfrak w}\) and an eigenvalue \(\lambda_j(Q(x))\) such that \(|\mu- \lambda_j(Q(x))|\leq z(F)\). In addition, if \(\beta(A)= \text{sup Re}(\sigma(A))\), \(r(A)\) is the spectral radius, then \(\beta(A)\leq \sup_{x\in{\mathfrak w}} \beta(Q(x))+ z(F)\), \(r(A)\leq \sup_{x\in{\mathfrak w}} r(Q(x))+ z(F)\).