Spectral Characterizations of Schatten-class perturbations of Partial isometries

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 $T^{\ast }T$, 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 ${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s}}(T^{\ast }T)$ is a singleton and the discrete spectrum of $T^{\ast }T$ satisfies a corresponding ${\mathrm{\ell }}^p$-summability condition. We further obtain equivalent criteria involving the compactness (or Schatten-class membership) of $\alpha I-T^{\ast }T$ and $\alpha T^{†}-T^{\ast }$. 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 Şerban and Turcu and obtain an explicit formula for the factorizing operator.