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Saturday, December 07, 2019

Of Hilbert space and vectors

In 1969, Vashishtha Narayan Singh published his PhD paper ‘Reproducing Kernels and Operators with a Cyclic Vector’. An overview of the dissertation that was supervised by John L Kelley

Updated: November 30, 2019 11:30:18 pm
Vashishtha Narayan Singh, Dr. Vashishtha Narayan Singh, Mathematician Vashishtha Narayan Singh, Vashishtha Narayan Singh Mathematician, India news, Indian Express Dr Singh’s paper seems to study the structure of a (relatively simple) class of operators.

(Written by Debashish Goswami)

Dr Vashishtha Narayan Singh’s dissertation paper, the only one he published on that problem, is about the structure of a certain class of bounded linear operators on Hilbert spaces, which is in the sub-area called ‘Operator Theory’ of maths.

A Hilbert space is a (possibly infinite dimensional) mathematical structure (a set with some operations) whose elements (called vectors) can be added, dilated or scaled by some number and there are also notions of some kind of ‘angles’ and distance between such vectors, including a ‘zero element’ which does not change any other element by addition.

This is a generalisation of a two-dimensional plane or three-dimensional space. (Linear) Operators on such spaces can be roughly thought of as some kind of infinite matrices. Recall that a (finite) matrix is nothing but a rectangular array of numbers arising typically to solve simultaneous linear equations in more than one unknown variable. For example, given the price of 2 kg rice + 3 kg wheat and also 3 kg rice + 5 kg wheat, if we want to find the prices of rice and wheat per kg, we will need to write a 2 by 2 matrix and solve the corresponding equations. Operators on Hilbert spaces turn out to be very important in physics, specially in quantum mechanical models and many other areas of physical and social sciences.

The theory of operators is thus an important branch and Dr Singh’s paper seems to study the structure of a (relatively simple) class of operators. It was a decent paper for a PhD student but it has only one citation, which means it did not really make any substantial breakthrough. It was published in a standard journal, the Pacific Journal of Mathematics.

I did like the writing style and clarity of arguments, which is quite commendable for a first paper by a very young student. It seems that the author did not continue the study of such operators afterwards. The work might have been of some relevance at that time, but the subject has advanced a lot since then and now the results obtained in that paper are unlikely to get much attention.

The writer is a professor at the Indian Statistical Institute, Kolkata

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