# Journal of Operator Theory

Volume 88, Issue 2, Fall 2022 pp. 275-288.

Trace formula for contractions and its representation in $\mathbb{D}$

**Authors**:
Arup Chattopadhyay (1), Kalyan B. Sinha (2)

**Author institution:**(1) Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, 781039, India

(2) Theoretical Sciences Unit, Jawaharlal Nahru Centre For Advanced Scientific Research, Bangalore, 560064, India

**Summary: **The aim of this article is twofold: give a short proof of the existence of real spectral shift function and the associated trace formula for a pair of contractions, the difference of which is trace-class and one of the two a strict contraction, so that the set of assumptions is minimal in comparison to those in all the existing proofs. The second one is to find a trace formula for differences of functions of a contraction and its adjoint, in which case, the integral in the formula is over the unit disc and has an expression surprisingly similar to the Helton--Howe formula.

**DOI: **http://dx.doi.org/10.7900/jot.2021may13.2323

**Keywords: **Kre\u \i n's trace formula, spectral shift function, self adjoint operators, unitary operators, contractions, unitary dilations

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