Perturbations of regularized determinants of operators in a Banach space (Q355698)

Let \(X\) be a Banach space with some additional properties. The author denotes by \(\Gamma_p\) a quasinormed ideal of compact operators in \(X\) endowed with the quasinorm \(N_{\Gamma_p}(\cdot)\). It is also assumed that \(N_{\Gamma_p}(\cdot)\) satisfies the property NEWLINE\[NEWLINE N_{\Gamma_p}(A+B)\leq b(\Gamma_p)(N_{\Gamma_p}(A)+ N_{\Gamma_p}(B)),NEWLINE\]NEWLINE where \(b(\Gamma_p)\) is a constant independent of \(A,B\in \Gamma_p\). The author proves the following two inequalities: NEWLINE\[NEWLINE |\det_p(I-A)|\leq \exp(a_p\gamma_p N_{\Gamma_p}^p (A)), NEWLINE\]NEWLINE and NEWLINE\[NEWLINE |\det_p(I-A)-\det_p(I-B)|\leq N_{\Gamma_p}(A-B)\exp [a_p\gamma_p b(\Gamma_p) (1+ \frac{1}{2}[ N_{\Gamma_p}(A-B)+N_{\Gamma_p}(A+B) ])^p], NEWLINE\]NEWLINENEWLINENEWLINEwhere \(\gamma_p=\frac{p-1}{p}\), \(a_p\) is a constant such that \(\sum\limits_{k=1}^\infty |\lambda_k(A)|^p\leq a_pN_{\Gamma_p}^p(A)\), the \(\lambda_k(A)\)'s (\(k=1,2,\dots\)) are the eigenvalues of the operator \(A\) counted with their algebraic multiplicities and \(\det_p(I-A)=\prod\limits_{k=1}^\infty E_p (\lambda_k(A))\), \(E_p\) being the Weierstrass primary factor (\(E_1(z)=(1-z)\), \(E_p(z)=(1-z)\exp \left(\sum\limits_{m=1}^{p-1}\frac{z^m}{m}\right)\), \(z\in \mathbb{C}\), \(p=2,3,\dots\)). As consequences of the main result, particular similar inequalities are obtained for \(p\)-summing operators, Hille-Tamarkin operators, and matrices and Lorentz ideal of compact operators.