Spectral Characterizations of Schatten-class perturbations of Partial isometries

Abstract

We characterize bounded operators that are compact (respectively, Schatten-class) perturbations of scalar multiples of partial isometries with finite-dimensional kernel. Our characterizations are formulated in terms of the essential spectrum of $T^*T$, absolutely norm attaining operators, and the Moore-Penrose inverse. In particular, we show that an operator $T$ is a Schatten-class perturbation of a partial isometry with finite-dimensional kernel if and only if $\sigma_{\mathrm{ess}}(T^*T)$ is a singleton and the discrete spectrum of $T^*T$ satisfies a corresponding $\ell^p$-summability condition. We further obtain equivalent criteria involving the compactness (or Schatten-class membership) of $\alpha I-T^*T$ and $\alpha T^\dagger-T^*$. As applications, we establish characterizations of compact and Schatten-class perturbations of isometries, describe the corresponding behavior of Moore--Penrose inverses, and derive factorization results for closed-range operators. In particular, we provide a new Moore--Penrose inverse proof of a theorem of \c{S}erban and Turcu and obtain an explicit formula for the factorizing operator.