[Paper Review] Spectra of Upper-triangular Operator Matrix

Shifang Zhang, Huaijie Zhong+1 more|arXiv (Cornell University)|Jun 27, 2009

Holomorphic and Operator Theory25 references3 citations

TL;DR

This paper investigates the spectral properties of 2×2 upper-triangular operator matrices $ M_C = \begin{pmatrix} A & C \\ 0 & B \end{pmatrix} $ on Banach spaces $ X \oplus Y $, analyzing 20 spectra and their filling-in-hole properties. It establishes that certain spectral inclusions—previously conjectured—do not hold, providing counterexamples that challenge existing assumptions in spectral theory of operator matrices.

ABSTRACT

Let $X$ and $Y$ be Banach spaces, $A\in B(X)$, $B\in B(Y)$, $C\in B(Y, X)$, $M_{C}=({cc}A&C 0&B)$ be the operator matrix acting on the Banach space $X\oplus Y$. In this paper, we give out 20 kind spectra structure of $M_C$, decide 18 kind spectra filling-in-hole properties of $M_C$, and present 10 examples to show that some conclusions about the spectra structure or filling-in-hole properties of $M_C$ are not true.

Motivation & Objective

• To systematically analyze the 20 distinct spectra associated with upper-triangular operator matrices $ M_C $ on $ X \oplus Y $.
• To determine which of these spectra satisfy the filling-in-hole property, particularly in relation to the spectra of the diagonal blocks $ A $ and $ B $.
• To challenge and correct previously claimed spectral inclusions, especially those involving essential and Weyl spectra.
• To provide explicit counterexamples demonstrating that certain spectral inclusions—such as $ \sigma_w(M_C) \subseteq \sigma_e(M_C) $—do not hold in general.

Proposed method

• The authors define 20 spectra for $ M_C $ using standard operator classes: invertible, Fredholm, Weyl, Browder, and their left/right variants.
• They analyze the filling-in-hole property by comparing the spectrum of $ M_C $ with the union of spectra of $ A $ and $ B $, particularly focusing on holes in the essential and Weyl spectra.
• Using functional analytic techniques, including ascent, descent, and index theory, they characterize the spectral sets via kernel and range properties of $ M_C - \lambda I $.
• They construct explicit examples in $ \ell^2 $ using shift operators $ T, S $, and auxiliary operators $ C_1, C_2, C_3 $ to test spectral inclusions.
• Counterexamples are built using specific operators: $ A = T $, $ B = S $, $ C = C_3 = I - TS $, and variations with $ T_1, S_1 $, and block matrices.
• The analysis includes checking spectral inclusions via $ \sigma_{se}(M_C) \stackrel{?}{=} \sigma_{se}(A) \cup \sigma_{se}(B) \cup (\overline{\sigma_p(A^*)} \cap \sigma_p(B)) $, showing this equality fails.

Experimental results

Research questions

• RQ1 Does the essential spectrum of $ M_C $ satisfy $ \sigma_{se}(M_C) = \sigma_{se}(A) \cup \sigma_{se}(B) \cup (\overline{\sigma_p(A^*)} \cap \sigma_p(B)) $?
• RQ2 Is the Weyl spectrum of $ M_C $ always contained in the essential spectrum of $ M_C $?
• RQ3 Do the spectra $ \sigma_{rD} $ and $ \sigma_{lD} $ satisfy the generalized filling-in-hole property with respect to $ \sigma_{rD}(A) \cup \sigma_{rD}(B) $?
• RQ4 Can the inclusion $ \sigma_w(M_C) \subseteq \sigma_e(M_C) $ fail for some $ C \in B(Y,X) $?
• RQ5 Is the inclusion $ \sigma_D(M_C) \subseteq \sigma_e(M_C) $ always true, or can it be strict?

Key findings

• The inclusion $ \sigma_w(M_C) \subseteq \sigma_e(M_C) $ does not hold in general; a counterexample is constructed with $ A = T $, $ B = S $, $ C = C_3 $.
• The inclusion $ \sigma_w(M_C) \subseteq \sigma_D(M_C) $ fails when $ A = T $, $ B = S $, $ C = 0 $, showing $ \sigma_w $ is not contained in $ \sigma_D $.
• The inclusion $ \sigma_e(M_C) \subseteq \sigma_D(M_C) $ is false; a counterexample uses $ A = T_1 $, $ B = \begin{pmatrix} S_1 & 0 \\ 0 & S \end{pmatrix} $, $ C = (C_1, 0) $.
• The inclusion $ \sigma_e(M_C) \subseteq \sigma_w(M_C) $ fails when $ A = T_1 $, $ B = S_1 $, $ C = C_2 $, demonstrating that $ \sigma_e $ is not contained in $ \sigma_w $.
• The inclusion $ \sigma_D(M_C) \subseteq \sigma_e(M_C) $ fails when $ A = T $, $ B = \begin{pmatrix} S & 0 \\ 0 & 0 \end{pmatrix} $, $ C = (C_3, 0) $, and $ \sigma_D(M_C) $ is not contained in $ \sigma_e(M_C) $ or $ \sigma_w(M_C) $.
• The claimed spectral identity $ \sigma_{se}(M_C) = \sigma_{se}(A) \cup \sigma_{se}(B) \cup (\overline{\sigma_p(A^*)} \cap \sigma_p(B)) $ is invalid; a counterexample shows the difference set $ W $ is a single point, not an open set.