Introduction to Fredholm, Toeplitz and Hankel Operator

Roy, Subhajit and D, Sukumar (2019) Introduction to Fredholm, Toeplitz and Hankel Operator. Masters thesis, Indian institute of technology Hyderabad.

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Abstract

The whole discussion of the thesis is about Fredholm; Toeplitz; and Hankel operator. Atkinson0s theorem gives an equivalent de_nition of Fredholm operator, there we get a relation between Compact and Fredholm operator. We will learn the Calkin algebra on a Hilbert space and what is the relation between the Fredholm operator and this algebra, the index of a Fredholm operator. The sum of Compact and Fredholm operator is Fredholm and the index is the same as the index of the Fredholm operator. The characteristic of Fredholm operator of index 0 and we will show that the index map is a locally constant map. We will learn about the Hardy space and Toeplitz operator on Hardy space. All the Toeplitz operator on the Hardy space is not Fredholm, there is a class of Toeplitz operator which are Fredholm and the Toeplitz operator which is compact is only the zero operators. Also, we will learn about some properties about the set of all Toeplitz operator. Finally, we will learn Hankel operator on Hardy space and some algebraic properties of the Hankel operator. In general, all Hankel does not commute with a Toeplitz operator, if a Hankel commute with a symmetric Toeplitz operator, then the Toeplitz operator is a constant multiple of the identity operator.

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