# Journal of Operator Theory

Volume 91, Issue 1, Winter 2024 pp. 261-294.

Pairs of projections and commuting isometries

**Authors**:
Sandipan De (1), Jaydeb Sarkar (2), P. Shankar (3), T.R. Sankar (4)

**Author institution:** (1) School of Mathematics and Computer Science, Indian Institute of Technology Goa, Farmagudi, Ponda-403401, Goa, India

(2) Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, 560059, India

(3) Department of Mathematics, Cochin University of Science and Technology, Kochi 682022, Kerala, India

(4) Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400076,
India

**Summary: ** It is known that the non-zero part of compact defect operators of Berger-Coburn-Lebow pairs (BCL pairs in short) of isometries are diagonal operators of the form
$\left\{
\begin{smallmatrix}
I_1 & & & \\
& D & & \\
& & - I_2 & \\
& & & - D \\
\end{smallmatrix}
\right\},
$
where $I_1$ and $I_2$ are the identity operators and $D$ is a positive contractive diagonal operator. We discuss the question of constructing an irreducible BCL pair from a diagonal operator of the above type. The answer to this question is sometimes in the affirmative and sometimes in the negative. This also answers a part of the question raised by He, Qin, and Yang. Our explicit constructions of BCL pairs yield concrete examples of pairs of commuting isometries.

**DOI: **http://dx.doi.org/10.7900/jot.2022apr19.2370

**Keywords: ** shift operators, isometries, projections, weighted shifts, weighted shift matrices, Toeplitz operators, compact operators

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