Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)


Ghosh, Jayanta Kumar (TSMU-Kolkata; ISI)

Abstract (Summary of the Work)

Several articles have appeared in recent years which discuss linear transformations on finite-dimensional vector spaces over the quaternions in terms of matrices, but the general infinite -dimensional situation does not seen to have received much attention. In particular, very little is known about linear transformations on quaternionic Hilbert spaces apart from the obvious theory of Hermitian operators. There are hardly any discussion of this subject, apart from the brief treatment of Finkelstein, Jauch, Schiminovitch and Speiser in their fundamental paper on the foundations of quaternion quantum mechanics (1962), which gives spectral theorems for unitary operators and skew hermitian operators and a forn of Stone's theorem which says that weakly continuous one-parameter unitary groups on quaternionic Hilbert spaces have skew hermitian infinitesnimal generators, Varadarajan's book on the geometry of quantum mechanics (1968) in large parts of which Hilbert spaces over the reals, complex numbers and quaternions are discussed simultaneously. and Each's article (Each, 1963) on quaternionic quantum mechanics.We shall now make a thorough study of the central area of problems featuring normal operators and their structure. The first four chapters introduce quaternionic Hilbert spaces and develop a neat and powerful technique of handling normal operators on each spaces and the next four chapters exploit this theory to yield not only the complete structure theory of the individual normal operator but also an understanding of the structure of one-parameter unitary groups and weakly closed abelian algebras of operators. In the treatment of the ac problems the emphasis is on departures from the complex case and all proofs even remotely resembling proofs of similar theorems on complex Hilbert spaces (almost all of which may be found in the works of either Halmos (1957) or Segal (1951) or varadarajan (1959)) are omitted.A brief summary of the contents an follows:Chapter I serves to establish notational and other conventions and to collect together the elementary properties of the division ring Q of quaternions. The structure of closed division subrings of Q is then clarified and a classification of equivalence classes of continuous one-parameter groups of unit quaternions in terms of non-negative real numbers is deduced. This is used later in Chapter V where we seek an analogue of Stone's theorem on one -parameter unitary groups on quaternionic Hubert spaces.


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