Separation of spectra for block triangles

Abstract

Abstract. Separation of the spectra of the diagonal elements of a block triangle corresponds to comparison with its fundamental projection.is a Banach algebra with block structure: thus [3] A and B are Banach algebras with identities and M and N are bimodules over A and B and everything is explained by formal matrix multiplication. An upper triangle in G is an element of the formIt is well known [2], [3], that of the three spectraeach is contained in the union of the other two: this extends more generally to "spectral triangles" [4]. Disjointness between the spectra of a ∈ A and b ∈ B, or significant subsets of them, has consequences expressible [3] in terms of a comparison between the operatorsis holomorphic near the spectra of a ∈ A and b ∈ B, thenIn particular if and only if S has disjoint diagonal spectrathen P is a holomorphic function of S:(1.3) P = f (S) with f ∈ Holo σ(S) .

Simultaneous solutions of operator Sylvester equations

“…where X is the simultaneous solution of equations (1.3). Corollary 4.6 for the case of a single operator (k = 1) is contained in [4].…”