Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Research and Training School (RTS)


Nadkarni, Mahendra G. (RTS-Kolkata; ISI)

Abstract (Summary of the Work)

The theory of invariant subspaces of various Cunclion-spaces of complox-valued and vector-valued functions on the circle group 13 well known through the 'Loctures on Invariant Subspaces' by Helson ([4]), Replacin; Line circle croup by a Bohr group B (that is, a compact ahelian group whose dual is a subgroup of tho ronl. line R, dense in the topology of R), Helson and Lowdenalager initiated the study of invariant subspaces of L2,(B) in (6]. They discovered that after suitable normalisation, the simply invariant subspaces of L2(B) arcinonc-to-one correspondence with a certain class of functions on R x B, which are called cocycles, Further contributions came from many authors : Holson ([5); I, II), Gamolin ([2), ch. VII), Yale ((16]) and Helson and Kahano ([8]), Also the work of Helson and Lowdenslager on multivariate prediction ([7]; I, 11) has points of contact with this thery.In this dissertation we extend cortain parts of the above theory to invariant subspaces of L2-spaces of Hilboert spaco- valued functions under the simplifying assurption that has. n countable dual.. The Holson-Lowdenslager correspondence extends with the difference that the cocycles, now, nrc operator- valued. This is accomplished by resorting to the theory of systoms of imprimitivity which 1a extensively emplayed in representation theory of locally compact groups.The material is divided info three chapters, In the first chapter we go into the relatiotusihp of cocyeles with system: of imprimitivity for the special rase whore a second countable Iocally compact abolian group nets dousely Into another, This lends to ducl systems of imprimitivity which are not; available in the general case. Next, our metion of cocyeles is widor than usual ; we allow thom to have partial isometries as values, This permits us to descritbe system of imprimitivity which do not act in the who Le HIlbert space, We also obtain generalisation of the strueture of coyeles given by Gnmoljn (12), ch. VIT, sec, 11), We use this to discuss the irreducibility of certain systems of imprimitivity.In chapter II we come to invarinnt subapeces. Here most of the work consists in recomising me structares, to which, then, the results n chapter I readily apply. Tho principal correspondonce theorem between simply invariant subspaces and cocyclos appears in section 6. In the same section we show that there are simply invariant subspaces of the veetarial L2, which are not equivalent to a direct sum of simply invartant subspaces of the scalar L2. In section 8 we obtain an analytic description of simply invariant subspaces in terms of cocycles simllar to one given by Helson (L5), I) fer the scalar case. chapter III deals with an application of the results of chapter II, to prediction theory. We consider a multivariste stationary stochastic process with time Γ (a countable was subspaces of R). We find necossary and sufficient conditions for such n process to he purely on-deterministic.


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