Operators satisfying a similarity condition

ABSTRACT

Given Hilbert space operators $A,S\in B(\mathcal{H})$ such that $0\notin \overline{W(S)}$ ($=$ the closure of the numerical range of S), the similarities $AS{A}^{\ast }=S$ for invertible A and $AS=S{A}^{\ast }$ have been considered by a number of authors over past few decades. A classical result of C. R. De Prima (resp., I. H. Sheth) says that if A and $A^{-1}$ are normaloid or convexoid (resp., A is hyponormal), then $AS{A}^{\ast }=S$ implies A is unitary (resp., $AS=S{A}^{\ast }$ implies A is self-adjoint). This paper uses (Putnam–Fuglede theorem type) commutativity results to obtain generalizations of extant results on similarities of the above type. Amongst other results, it is proved that if $AS{A}^{\ast }=S$ with A invertible and $0\notin \overline{W(S)}$, then: (i) A normaloid implies either A is unitary or ${\sigma }_p(A)=\mathrm{\varnothing }$; (ii) operators A satisfying the positivity condition $|A^2{|}^2-2{|A|}^2+I\ge 0$ are unitary. If the operator A in $AS{A}^{\ast }=S$ (resp., $AS=S{A}^{\ast }$) is w-hyponormal or class $\mathcal{A}(1,1)$ with $A^{-1}(0)\subseteq {A^{\ast }}^{-1}(0)$, then a sufficient condition for A to be unitary (resp., A to be self-adjoint) is that $0\notin W(X)$; furthermore, one may drop the hypothesis $A^{-1}(0)\subseteq {A^{\ast }}^{-1}(0)$ in the case in which $0\notin \overline{W(X)}$.

KEYWORDS:

• Bounded and unbounded Hilbert space operators
• numerical range
• SVEP
• commutativity theorem
• self-adjoint operator
• unitary operator

AMS SUBJECT CLASSIFICATIONS:

• Primary: 47A05
• 47A62
• 47A12
• 47B47

Acknowledgements

The authors thank a referee for his very helpful comments.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of the second author was supported by a grant from the National Research Foundation of Korea (NRF), funded by the Korean government [grant number NRF-2016R1D1A1B03930744].