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Fredholm spectra of £f-Toeplitz operatorsChen, Chih-Hao 25 July 2011 (has links)
Abstract
Let £f be a complex number in the closed unit disc , and H be a separable Hilbert space with the orthonormal basis, say,£`= {en : n =0 , 1 , 2¡K}. A bounded operator T on H is called a £f-Toeplitz operator if <Tem+1 , en+1> =£f <Tem , en> (where <¡E,¡E> is the inner product on H).If the function £p can be represented as a linear combination of the above orthonormal basis with the coefficients an=<Te0 ,en >, n≥ 0,and an=<Telnl ,e0 >, n<0, then we call this the symbol of T . The subject arises naturally from a special case of the operator equation
S*AS =£fA + B; where S is a shift on H ,
and in this operator equation the matrix A can solve a special set of simultaneous equations.
It is also clear that the well-known Toeplitz operators are precisely the solutions of S*AS = A, when S is the unilateral shift.In this paper,we will review the similarities and differences between £f-Toeplitz operators and Toeplitz operators. The main purpose is to generalize the well-known Coburn's characterization for the essential spectrum(or,the same in this case,spectrum)for Toeplitz operators to £f-Toeplitz operators.
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