This note is devoted to the spectral analysis of rank one perturbations of unitary and self-adjoint operators. We study the following question: given two cyclic (i.e., having simple spectrum) operators A and B, when is A equivalent to B up to a rank one perturbation? More precisely, when does there exist a unitary operator U such that rank (UAU∗−B) = 1? As usual, we are looking for an answer in...

The property of cyclicity of a linear operator, or equivalently the property of simplicity of its spectrum, is an important spectral characteristic that appears in many problems of functional analysis and applications to mathematical physics. In this paper we study cyclicity in the context of rank-one perturbation problems for self-adjoint and unitary operators. We show that for a fixed non-zer...

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We study the eigenvalue trajectories of a time dependent matrix Gt=H+itvv∗ for t≥0, where H is an N×N Hermitian random and v unit vector. In particular, we establish that with high probability, outlier can be distinguished at all times t>1+N−1∕3+ε, any ε>0. The this natural process combines elements non-Hermitian analysis, illustrates some aspects intrinsic instability (even weakly) matrices.