2026/07/16 by Neeru Bala, Ramesh Golla
#math.FA
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 σess(T^*T) is a singleton and the discrete spectrum of T^*T satisfies a corresponding ℓp-summability condition. We further obtain equivalent criteria involving the compactness (or Schatten-class membership) of αI-T^*T and αT^†-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 Şerban and Turcu and obtain an explicit formula for the factorizing operator.