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On the fine spectrum of the operator B(r,s,t) over the sequence spaces 1 and bv. (English) Zbl 1152.47024

Let A=(a nk ) n,k be an infinite matrix. For a complex sequence x=(x k ) k , let Ax be, formally, the sequence with coefficients (Ax) n := k a nk x k . For any complex numbers r, s and t (with s and t not simultaneously null), let A=B(r,s,t) be the infinite matrix with a nn =s, a n+1,n =s and a n+2,n =t (n) and a nk =0 otherwise. It is well-known that B(r,s,t) defines a bounded linear operator over 1 and b v with B(r,s,t) 1 orb v =|r|+|s|+|t|.

The paper under review deals with spectral properties of this operator over 1 and b v . In particular, the authors show that the residual spectrum σ r (B(r,s,t)) and the usual spectrum σ(B(r,s,t)) of B(r,s,t) over 1 or b v coincide and are equal to

S:=α:2(r-α) s 2 +s 2 -4t(r-α)1,

so the point (discrete) spectrum σ p (B(r,s,t)) and the continuous spectrum σ c (B(r,s,t)) are empty (here, for a complex value z, z will denote the unique square root of z with principal argument in [0,π)).

Some results of this paper extend other ones by H. Furkan, H. Bilgiç and K. Kayaduman [Hokkaido Math. J. 35, No. 4, 893–904 (2006; Zbl 1119.47005)] and H. Furkan and K. Kayaduman [Int. Math. Forum 1, No. 21–24, 1153–1160 (2006; Zbl 1119.47306)].

47B37Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
40C05Matrix methods in summability
47A10Spectrum and resolvent of linear operators