On Hankel operators associated with a class of non-Toeplitz operators
The author considers operators on the space of all square-summable complex sequences. \((e_ n)\) denotes the standard basis. If \(\phi\) is some essentially bounded function on the unit circle having Fourier coefficients \((c_ k)\) the associated Toeplitz operator is that having the following matrix with respect to the standard basis. Set \(c_ 0\) constantly on the main diagonal, \(c_ 1\) on the first parallel to the main diagonal situated under this one, \(c_{-1}\) on the first parallel above the main diagonal and so on. The corresponding Hankel operator is associated to the Toeplitz operator in the usual way. Denote by S the classical shift operator of multiplicity 1. To any operator T the author associates the sequence \(H_ n(T)=J_ nTS^{n+1}\) where \(J_ ne_ i=e_{n-i}\) for \(0\leq i\leq n\) and \(J_ ne_ i=0\) if \(i>0\). The weak limit of the sequence \(H_ n(T)\) if it exists is denoted H(T) and is a Hankel operator. If T is Toeplitz then H(T) is the corresponding Hankel operator. A necessary and sufficient condition for the existence of H(T) is given. The existence of H(T) for a large class of non-Toeplitz operators is deduced. Operator theoretic analogs for classical results connecting Toeplitz operators to function theory are given.
scientific article; zbMATH DE number 4208881
scientific article