Closed operators in Hilbert space defined by a non-self-adjoint resolution of the identity $\{X(\lambda)\}_{\lambda\in {\mb R}}$, whose adjoints constitute also a resolution of the identity, are studied . In particular, it is shown that a closed operator $B$ has a spectral representation analogous to the familiar one for self-adjoint operators if and only if $B=TAT^{-1}$ where $A$ is self-adjoint and $T$ is a bounded operator with bounded inverse.
Inoue, A., Trapani, C. (2014). Non-Self-Adjoint Resolutions of the Identity and Associated Operators. COMPLEX ANALYSIS AND OPERATOR THEORY, 8(7), 1531-1546 [10.1007/s11785-014-0359-1].
Non-Self-Adjoint Resolutions of the Identity and Associated Operators
TRAPANI, Camillo
2014-01-01
Abstract
Closed operators in Hilbert space defined by a non-self-adjoint resolution of the identity $\{X(\lambda)\}_{\lambda\in {\mb R}}$, whose adjoints constitute also a resolution of the identity, are studied . In particular, it is shown that a closed operator $B$ has a spectral representation analogous to the familiar one for self-adjoint operators if and only if $B=TAT^{-1}$ where $A$ is self-adjoint and $T$ is a bounded operator with bounded inverse.
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