Conjugating by singular operators: On the boundedness of similarity transforms near singular points

2024/10/24 by Daniel Falkowski, Falkowski, Daniel, Carl-Fredrik Lidgren +1
Mathematics · #advanced mathematical theories #Mathematical Analysis and Transform Methods #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.2410.19161

Abstract

We consider the question of, given operators A, Z and a sequence of invertible operators U_n→ Z, whether the sequence U_nAU_n^-1 is bounded in norm, as well as generalizations of this where U_nAU_n^-1 is modified by some bounded linear map on bounded linear operators. In the setting of Hilbert spaces, we provide a complete classification in terms of algebraic criteria of those A for which such a sequence exists, as long as Z is of generalized index zero, which always holds in finite-dimensional contexts. In the process, we prove that particular coefficients arising in inverses of certain good paths going to Z can also be classified in terms of an entirely algebraic criterion.

Related

• Similar transformation of one class of correct restrictions
• Full operators and algebras generated by invertible operators
• Operators Whose Conjugation Orbits Satisfy Polynomial Growth Conditions
• Operators near completely polynomially dominated ones and similarity problems
• A similarity invariant of a class of n-normal operators in terms of K-theory