arXiv:2105.02134v3 [math.FA] 17 May 2022 THE JOINT SPECTRUM FOR A COMMUTING PAIR OF ISOMETRIES IN CERTAIN CASES TIRTHANKAR BHATTACHARYYA, SHUBHAM RASTOGI, AND VIJAYA KUMAR U. In fond memory of J¨ org Eschmeier Abstract. We show that the joint spectrum of two commuting isometries can vary widely depending on various factors. It can range from being small (of measure zero or an analytic disc for example) to the full bidisc. En route, we discover a new model pair in the negative defect case and relate it to the modified bi-shift. 1. Introduction An isometry V is called pure if V ∗n converges to 0 strongly as n →∞. This is equivalent to saying that V is the unilateral shift of multiplicity equal to the dimension of the range of the defect operator I − VV ∗ . The famous Wold decomposition [17, 23] tells us that given an isometry V on a Hilbert space H, the space H breaks uniquely into a direct sum H = H 0 ⊕H ⊥ 0 of reducing subspaces such that V | H 0 is a unitary and V | H ⊥ 0 is a pure isometry. This immediately implies that for a non-unitary isometry V (i.e., when the defect operator is positive and not zero), the spectrum σ(V ) is the closed unit disc D = {z ∈ C : |z|≤ 1}. The situation for a pair of commuting isometries is vastly different. The topic of commuting isometries has been vigorously pursued in the last two decades, see [1, 2, 4, 5, 6, 11, 15, 16, 18, 20, 21] and the references therein. In [6] and [5], the novel idea of using graphs has led to a clear understanding of structures. The defect operator C (V 1 ,V 2 ) is introduced in [12] and [13], as C (V 1 ,V 2 )= I − V 1 V ∗ 1 − V 2 V ∗ 2 + V 1 V 2 V ∗ 2 V ∗ 1 . In [13] and [16], the authors provide the characterization of (V 1 ,V 2 ), when the defect is positive, negative or zero. It is well known (see [13, 11]) that a pair has positive defect if and only if it is doubly commuting, and it has negative defect if and only if it is dual doubly commuting. In all the three cases, the defect is either a projection or negative of a projection. In general the defect is the difference of two projections; see [16]. In this paper we study the pairs of commuting isometries, whose defect is the difference of two mutually orthogonal projections. We characterize such pairs in Theorem 2.1 and we classify them in Table 1. We also provide the characterization for a few cases in Table 1; see Lemma 6.3 and Lemma 6.9. We rephrase the structure of (V 1 ,V 2 ) in each case, which appears in Table 1, in a unified approach using the Berger-Coburn-Lebow (BCL) Theorem. The joint spectrum is studied in detail for all the cases except the last one appearing in Table 1. MSC: Primary: 47A13, 47A45, 47A65. 1