A note on compact operators and operator matrices. (English) Zbl 0938.47019
Summary: Two properties of compact operators acting on a separable Hilbert space are discussed. In the first part a characterization of compact operators is obtained for bounded operators represented as tri-block diagonal matrices with finite blocks. It is also proved that one can obtain such a tri-block diagonal matrix representation for each bounded operator starting from any orthonormal basis of the underlying Hilbert space by an arbitrary small Hilbert-Schmidt perturbation.
The second part is devoted to the so-called Hummel’s property of compact operators: each compact operator has a uniformly small orthonormal basis for the underlying Hilbert space. The class of all bounded operators satisfying Hummel’s condition is determined.
The second part is devoted to the so-called Hummel’s property of compact operators: each compact operator has a uniformly small orthonormal basis for the underlying Hilbert space. The class of all bounded operators satisfying Hummel’s condition is determined.
MSC:
47B07 | Linear operators defined by compactness properties |
47A10 | Spectrum, resolvent |
47A66 | Quasitriangular and nonquasitriangular, quasidiagonal and nonquasidiagonal linear operators |
47A55 | Perturbation theory of linear operators |
46B15 | Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces |