Optimal Weyl-type inequalities for operators in Banach spaces
Let \((s_n)\) be an \(s\)-number sequence. The authors show, for each \(k = 1, 2, \dots\) and \(n \geq k + 1\), the inequality \[ {\left( \prod _{j=1}^n | \lambda_j (T)| \right)}^{1/n}\leq (k+1)^{1/2}\| T\| ^{1-k/n} {\left( \prod^n _{j=n-k+1} s_j(T)\right)}^{1/n} \] between the eigenvalues and \(s\)-numbers of a compact operator \(T\) in a Banach space. The proof is given for the smallest scale of \(s\)-numbers, namely the Hilbert numbers, for which several other inequalities are shown to this purpose. Using Fourier matrices and diagonal matrices, the authors confirm that the constant \((k + 1)^{1/2}\) is optimal for \(n = k + 1\) and \(k = 1, 2,\dots\). They also give some hints why this inequality seems to be an appropriate tool for estimating the first single eigenvalues. On the other hand, the authors prove that the Weyl numbers form a minimal multiplicative \(s\)-number sequence in the following sense. Let \((s_n)\) be a multiplicative \(s\)-number sequence. Then, if for all linear and continuous operators \(T\) between Banach spaces it holds that \(s_n(T) \leq x_n(T)\), \(n=1,2, \dots\), it follows that \(x_{2n-1}(T) \leq e \, (\prod_{k=1}^{n} s_k(T))^{1/n}\) for \(n=1,2, \dots\).
scientific article; zbMATH DE number 5156940