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A matriceal analogue of Fejér’s theory. (English) Zbl 1043.15020

Let \(A=(a_{ij}), i,j\geq 0\) be an infinite matrix with complex entries and let \(n\geq 0.\) The matrix \(A\) is called of \(n\)-band type if \(a_{ij}=0\) for \(| i-j| >n.\) For \(k=0,\pm 1,\pm 2,\cdots ,\) let \(A_k=(a'_{ij}),\) where \(a'_{ij}=a_{ij}\) if \(j-i=k;\) \(a_{ij}'=0,\) otherwise. \(A_k\) is called the Fourier coefficient of the matrix \(A.\) Let \(B(\ell_2)\) be the space of bounded linear operators on the one-sided scalar \(\ell_2\)-space and \(A\) be a matrix corresponding to an operator on \(B(\ell_2).\) Analogously to Fejér’s theory for Fourier series, the authors define Cesaro sum by \[ \sigma_n(A)=\sum_{k=-n}^{k=n}A_k\left (1-\frac{| k| }{n+1}\right ). \] Then they say that \(A\) is a continuous matrix if \(\lim_{n\rightarrow \infty }\| \sigma _n(A)-A\| _{B(\ell_2)}=0.\) In this way the authors extend the classical Banach space of functions \(C(T)\) on the torus \(T\) to the space of continuous matrices. The space of all continuous matrices is denoted by \(C(\ell_2).\) In Section 3, the authors derive some properties and relations between the spaces \(B(\ell_2)\) and \(C(\ell_2).\) Similar extensions are considered for the space \(L^1(T).\) Theorem 4.2, Theorem 4.3 and Theorem 4.10 are the main results.

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
42A45 Multipliers in one variable harmonic analysis
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
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