Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name

Computer Science


Theoretical Statistics and Mathematics Unit (TSMU-Delhi)


Sarkar, Jaydeb (TSMU-Delhi; ISI)

Abstract (Summary of the Work)

A very general and fundamental problem in the theory of bounded linear operators on Hilbert spaces is to find invariants and representations of commuting families of isometries.In the case of single isometries this question has a complete and explicit answer: If V is an isometry on a Hilbert space ℋ, then there exists a Hilbert space Hu and a unitary operator U on ℋu such that V on ℋu and[ S ⊗ IW 0 0 U] ∈ B((l 2 (ℤ+) ⊗ W) ⊕ ℋu),are unitarily equivalent, whereW = ker V∗ ,is the wandering subspace for V and S is the shift operator on l 2 (ℤ+) [66]. This fundamental result is due to J. von Neumann [81] and H. World [110] (see Theorem 1.2.1 for more details).In one hand, unitary operators are completely determined by the representing spectral measure. And, on the other hand, given n ∈ ℕ ∪ {∞}, there exists precisely one Hilbert space ε, up to unitary equivalence, of dimension n (here all Hilbert spaces are assumed to be separable), and given a Hilbert space ε, there exists precisely one shift operator, up to unitary equivalence, of multiplicity dim ε on some Hilbert space ℋ. Therefore, multiplicity is the only (numerical) invariant of a shift operator. Note that shift operators are special class of isometries, and moreover, the defect operator of a shift determines the multiplicity of the shift.Now we turn to tuples of commuting isometries on Hilbert spaces. It is remarkable that tractable invariants (whatever it means including the possibilities of numerical and analytical invariants) of commuting pairs of isometries are largely unknown. We stress on the fact that the case of pairs of commuting isometries itself is more subtle, and is directly related to the commutant lifting theorem [51] (in terms of an explicit, and then unique solution), invariant subspace problem [70] and representations of contractions on Hilbert spaces in function Hilbert spaces [79]. For instance:(a) Let S be a closed joint (Mz1 , Mz2 )-invariant subspace of H2 (D 2 ), the Hardy space over the bidisc ⅅD2 . Then (Mz1 |S, Mz2 |S) on S is a pure (see Chapter 3) pair of commutingisometries. Classification of such pairs of isometries is largely unknown (see Rudin [94, 93]).(b) Let T be a contraction on a Hilbert space ℋ. Then there exists a pair of commuting isometries (V1, V2) on a Hilbert space K such that T and Pker V ∗ 2 V1|ker V∗ 2 are unitarily equivalent (see Bercovici, Douglas and Foias [18]).(c) The celebrated Ando dilation theorem (see Ando [9]) states that a commuting pair of contractions dilates to a commuting pair of isometries. Again, the structure of Ando’s pairs of commuting isometries is largely unknown.(d) Contrary to the simpler structure of shift invariant subspaces of the one variable Hardy space, structure of invariant subspaces for (Mz1 , . . . , Mzn ) on H2 (Dn ), n > 1, is quite complicated. For example (see Rudin [94, 93]): There exist invariant subspaces S1 and S2 for (Mz1 , Mz2 ) on H2 (D2 ) such that (i) S1 is not finitely generated, and (ii) S2 ∩ H∞(D2 ) = {0}.


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