On approximation spectrum of bounded self-adjoint operators
Let T be a bounded linear operator on the Hilbert space \(\ell_ 2\) with the standard basis \(e_ k=\{\delta_{j,k}\}^{\infty}_{j=1}\), \(k=1,2,..\). and \(H_ n\) denotes the n-dimensional subspace spanned by \(\{e_ 1,e_ 2,...,e_ n\}\). The author says that a complex number \(\lambda\) belongs to the approximation spectrum \(\sigma_ a(T)\) of T if \(\lim_{n\to \infty}\| (P_ nTP_ n-\lambda I)| H_ n)^{- 1}\| =+\infty\), where \(P_ n\) denotes the orthogonal projection from \(\ell_ 2\) on \(H_ n\). And he says that T is a \(2n+1\) diagonal operator if \(t_{i,j}=0\) for \(| i-j| >n\), where \(\{t_{i,j}\}^{\infty}_{i,j=1}\) is the matrix of T in the above fixed basis of \(\ell_ 2.\) If T is self-adjoint it is clear that \(\sigma (T)\subset \sigma_ a(T)\), where \(\sigma\) (T) denotes the spectrum of T. In this paper, the author considered the following problem: When does the equality \(\sigma_ a(T)=\sigma (T)\) holds? And he proved the following Theorem. Let T be a bounded linear self-adjoint five diagonal operator in \(\ell_ 2\), then \(\sigma_ a(T)=\sigma (T)\). There exists a seven diagonal bounded linear self-adjoint operator T' for which \(\sigma_ a(T')\neq \sigma (T')\).
scientific article; zbMATH DE number 3981850