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- W1991056476 abstract "Douglas's localisation theory for Toeplitz operators is used to show that there exist certain Hankel operators with discontinuous symbols which do not lie in the C*-algebra generated by the Toeplitz operators. As a simple corollary we also see that these operators do not lie in the closed linear span of the positive Hankel operators. Introduction. Let L be the Hilbert space of square integrable functions on the unit circle T, with complete orthonormal basis (en; n E Z) where en denotes the function z'. J is the unitary on L defined by Jen = e_n for n in Z. H2 denotes the Hardy subspace spanned by (en; n > 0} and P is the orthogonal projection of L2 onto H2. For an essentially bounded measurable function 4) in Le, the Toeplitz operator T7 on H 2, is defined by T-, = PM,IH2 where Mf is the usual multiplication operator on L. We call 4) the symbol of the Toeplitz operator T,. The Hankel operator on H2, with symbol 4 in LX, is defined by 5,0= PJM,IH2. Whilst the Toeplitz and Hankel operators are formally rather similar, quite different techniques are usually needed to study them. However in this note we shall use Douglas's localisation theory for Toeplitz operators [2] to show that the closed linear span of the positive Hankel operators does not contain certain Hankel operators with discontinuous symbol. This result has also appeared in [4]. The proof was based on Widom's characterisation of positive Hankel operators [5] which we avoid here. Notation. For 4 in L, the functions 4p* and + are defined by +* (z) = +(z) and +(z) = 0* (Z) (z E T). Thus we have JM, = M,*J. Let T denote the C*-algebra generated by the Toeplitz operators and let P denote the closed linear span of the positive Hankel operators. The following simple lemma shows that the symbol of the adjoint of a Received by the editors November 8, 1976. AMS (MOS) subject classifications (1970). Primary 47B35, 47C10." @default.
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- W1991056476 date "1977-01-01" @default.
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- W1991056476 title "Hankel operators with discontinuous symbol" @default.
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- W1991056476 doi "https://doi.org/10.1090/s0002-9939-1977-0512867-4" @default.
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