Spectrums of equivalent Schauder operators

TL;DR

This paper investigates how the Schauder spectrum of equivalent Schauder operators can differ significantly, especially under unitary transformations, revealing complex spectral behaviors related to the essential spectrum.

Contribution

It demonstrates that equivalent Schauder operators can have markedly different Schauder spectra, depending on the essential spectrum and unitary transformations.

Findings

Schauder spectrum can contain rings or circumferences depending on the essential spectrum.

Existence of unitary spread operators that alter the Schauder spectrum significantly.

Spectral properties are highly sensitive to unitary equivalence in Schauder operators.

Abstract

Assume that $T_1,T_2$ are equivalent Schauder operators. In this paper, we show that even in this case their Schauder spectrum may be very different in the view of operator theory. In fact, we get that if a self-adjoint Schauder operator $A$ has more than one points in its essential spectrum $\sigma_e(A)$, then there exists a unitary spread operator $U$ such that the Schauder spectrum $\sigma_S(UA)$ contains a ring which is depended by the essential spectrum; if there is only one point in $\sigma_e(A)$ and satisfies some conditions then there exists a unitary spread operator $U$ such that the Schauder spectrum $\sigma_S(UA)$ contains the circumference which is depended by the essential spectrum.