On properties of Crawford numbers of operators on Hilbert spaces

In this article, we conduct a comparative study of Crawford numbers and minimum moduli of bounded linear operators on Hilbert spaces. We provide a simple and distinct proof for the well-known equality of these two numbers for positive operators. Various estimates for Crawford numbers of operators have been derived, and the conditions under which an operator attains its Crawford number are discussed. We explore the relationships between subsets of the spectra and Crawford numbers of operators and also characterize the behaviour of Crawford numbers for self-adjoint operators. Additionally, we discuss the collection of operators whose Crawford numbers are zero, providing a detailed analysis and characterizing the conditions under which operators in this class attain their Crawford numbers.