Mathematica Moravica, Vol. 29, No. 2 (2025)

Michael Gil'
Bounds for the functional spectrum of operators in a Hilbert space
Mathematica Moravica, Vol. 29, No. 2 (2025), 1–11.
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Abstract. Let $\mathcal{H}_1$ and $\mathcal{H}_2$ be Hilbert spaces with the unit operators $I_1$ and $I_2$, respectively, and $A_{jk}$ be bounded operators acting from $\mathcal{H}_j$ into $\mathcal{H}_k$ $(j,k=1,2)$. We consider the block operator matrices $A=(A_{jk})$ and $F(z)={\rm diag} (\hat{K}_j(z)I_{j})$, where $\hat{K}_1(z)$ and $\hat{K}_2(z)$ are scalar analytic functions. The set of all $z\in\mathbb{C}$, such that $F(z)-A$ is boundedly invertible is called the $F$-regular set of $A$. The complement of the $F$-regular set to the complex plane is called the $F$-spectrum (the functional spectrum) of $A$. It is shown that the notion of the functional spectrum enables us to investigate, from the unified point of view, various types of coupled systems, in particular, systems of integral, fractional differential, integro-differential and differential-difference equations. We derive a bound for the functional spectrum and discuss applications of the obtained bound to the stability of the considered systems.
Keywords. Hilbert space, functional spectrum, operators, integral equations, fractional differential equations, stability.