| Summary: | 碩士 === 國立中山大學 === 應用數學系研究所 === 99 === Abstract
Let λ be a complex number in the closed unit disc , and H be a separable Hilbert space with the orthonormal basis, say,ε= {en : n =0 , 1 , 2…}. A bounded operator T on H is called a λ-Toeplitz operator if <Tem+1 , en+1> =λ <Tem , en> (where <‧,‧> is the inner product on H).If the function φ 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 =λA + 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 λ-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 λ-Toeplitz operators.
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