Spectral Structure

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98 3 Toeplitz Operators

(M )n 2 1 2

2 0 1
((ei ) )n(ei )2 d

2 En 1 ((ei ) )2 d

Also, n2 m(En). n 2 1 2

2 0 n(ei )2 d m(En) 0. Thus, if we dene fn n/ n , then {fn} is a sequence of unit vectors such that

Therefore (M). 1 (M )fn n. Now suppose / ess ran . Then there exists 0 such that m{ei : (ei ) } 0. This means that the function 1/( ) is dened almost everywhere, and, in fact, 1/ 1/ a.e. Thus 1/( ) L. But then the operator M 1

is bounded and is clearly the inverse of M . Thus / (M).

3.2 Basic Properties of Toeplitz Operators

The...

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