The residual spectrum and the continuous spectrum of upper triangular operator matrices

Abstract

Let H and K be separable infinite dimensional Hilbert spaces. We denote by M C the 2×2 upper triangular operator matrix acting on H ⊕ K of the form M C = (A C 0 B). For given operators A ∈ B(H) and B ∈ B(K), the sets ∪ C∈B(K ,H) σ r (M C) and ∪ C∈B(K ,H) σ c (M C) are characterized, where σ r (•) and σ c (•) denote the residual spectrum and the continuous spectrum, respectively.

The point spectrum and residual spectrum of upper triangular operator matrices

“…Remark 2.19. A description of the set X∈B(K ,H) σ r (M X ) was given in [13] (see (1)). From Theorem 2.12 and Proof.…”

“…with an unknown operator X ∈ B(K , H). See, e.g., [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. In [5,6,9,10,12,[14][15][16][17][18], the perturbations of different spectra (the spectra, left (right) spectra, point spectra, continuous spectra, residual spectra,• • • ) of M X were discussed.…”

“…were characterized, where σ lw (•) and Inv(K , H) denote the left Weyl spectrum and the set of all invertible operators from K into H. In [13], the set…”

Weyl type theorems of 2×2 upper triangular operator matrices

“…It is shown that for any T ∈ B(H), in [8,16,18,1], we get the implications: a-Weyl's theorem =⇒ Weyl's theorem =⇒ Browder's theorem, a-Weyl's theorem =⇒ a-Browder's theorem =⇒ Browder's theorem. [19,13].) Let A, B, C ∈ B(H).…”