On the Relationship Between the Pseudospectrum and the Quadratic Numerical Range for Upper Triangular Bounded Operator Matrices

This paper studies the pseudospectral inclusion property of upper triangular bounded operator matrices in Hilbert spaces. It is proven that the pseudospectra of upper triangular bounded operator matrices are contained within the closure of the quadratic numerical range. Our result extends the inclusion relationship between the spectra of block operator matrices and the quadratic numerical range, as well as the inclusion relationship between the pseudospectra and the numerical range, to the pseudospectral case. Under appropriate conditions, we characterize the distribution range of the pseudospectra of upper triangular bounded operator matrices, and provide an example to illustrate the validity of the conclusions.