## 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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