What question did this study set out to answer?

The aim is to characterize the weak approximate spectrum of upper triangular operator matrices and their relations to traditional spectra.

February 14, 2026Open Access

Some Fredholm left invertible completion problems for operator matrices

JDJiong DongChangzhi University

Key Points

The aim is to characterize the weak approximate spectrum of upper triangular operator matrices and their relations to traditional spectra.
Analysis of the operator matrix structure M_C in a Hilbert space setting.
Characterization of spectral relationships between sigma(M_C) and sigma_{Fa}(M_C).
Exploration of equivalent conditions for the weak property (C9_1) of the operator matrix.
Identified conditions under which sigma_{Fa}(M_C) equals sigma(M_C).
Described connections between the weak property (C9_1) and the spectra of the operators A and B in the matrix.

Abstract

The structure of the weak approximate spectrum of two-by-two upper triangular operator matrix MC = pmatrix A C \\ 0 B pmatrix acting on a Hilbert space H K is studied. First, we characterize the relationship between the spectrum (MC) and the weak approximate spectrum ₅₀ (MC) and give the equivalent conditions that make ₅₀ (MC) = (MC) and ₅₀ (MC) = ₅₀ (A) ₅₀ (B) according to the properties that the operators A and B satisfy. Then, we study the weak property (₁) of MC and explore the relationship between (MC) = ₅₀ (MC) and the weak property (₁) of MC.

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