Some Remarks on the Product Formula for Defect Numbers of Closed Operators

TL;DR

This work revisits Glazman's defect-number formula for products of closed operators and develops semi-Fredholm techniques to extend the defect-indices analysis to positive powers and real polynomials of a symmetric operator. It proves a product-index rule and shows that $n_{\pm}(P_m(S))=n_{\pm}(S^m)$ for real polynomials $P_m$, providing a unified framework for deficiency indices of operator powers. The authors then apply these abstract results to concrete differential operators, including Bessel-type and Legendre–type Sturm–Liouville problems, as well as PDE contexts like homogeneous Laplacian perturbations and singular Dirichlet Laplacians, deriving explicit deficiency indices such as $n_{\pm}((T_{Leg,min})^m)=2m$ and $n_{\pm}((T_{\Omega,h,k})^m)=m$. The paper highlights missed opportunities in cross-group communication and demonstrates the broad applicability of the defect-number formalism to both ODEs and PDEs.

Abstract

This largely pedagogical paper recalls some facts on defect numbers of products of closed operators employing results from the theory of semi-Fredholm operators and then applies these facts to positive integer powers of symmetric operators and subsequently to certain minimal Sturm--Liouville and minimal higher even-order ordinary and partial differential operators. We also point out some unexpected missed opportunities when comparing the work of different groups on this subject.

Theorems & Definitions (21)

• Lemma 2.2
• Definition 2.3
• Theorem 2.4
• Remark 2.5
• Theorem 2.6
• proof
• Remark 2.7
• Theorem 2.8
• proof
• Theorem 2.9