Commutants of some Hausdorff matrices

“…In papers [6] and [7] the author showed that there exist HausdorfT matrices with distinct diagonal entries that have non-Hausdorff matrices in the commutant, and that a necessary condition for the commutant of a HausdorfT matrix in T and A to be the same is that H have distinct diagonal entries.…”

Section: Introductionmentioning

confidence: 99%

“…Let ST C JH be the subalgebra consisting of lower triangular matrices. HausdorfT [7] showed that, if 77 is any HausdorfT matrix with distinct diagonal entries, then the commutant of H in ¿7" consists of HausdorfT matrices.Let T be the subalgebra of Jf consisting of all bounded infinite matrices on c, the space of convergent sequences, A the subalgebra of T composed of triangular matrices.In papers [6] and [7] the author showed that there exist HausdorfT matrices with distinct diagonal entries that have non-Hausdorff matrices in the commutant, and that a necessary condition for the commutant of a HausdorfT matrix in T and A to be the same is that H have distinct diagonal entries.The author [6] and Jakimovski [4] have independently shown that the cornmutant of (C, I), the Cesàro matrix of order 1, in T, is %?, the set of conservative HausdorfT matrices. (A matrix is conservative if it is convergencepreserving over c.) However, the proof only uses the fact that A, (C, 1 ) e 38 .…”

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confidence: 99%

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Commutants for Some Classes of Hausdorff Matrices

Rhoades¹

1995

Proceedings of the American Mathematical Society

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Abstract.Let T denote the algebra of all bounded infinite matrices on c , the space of convergent sequences, A the subalgebra of T consisting of lower triangular matrices. It is well known that, if H is any HausdorfT matrix with distinct diagonal entries, then the commutant of H in A contains only HausdorfT matrices. In previous work the author has shown that a necessary condition for the commutant of a HausdorfT matrix H to be the same in T and A is that H have distinct diagonal entries, but that the condition is not sufficient. In this paper it is shown that certain HausdorfT matrices, with distinct diagonal entries, have the same commutants in F and A .Let Jf be the set of all doubly infinite matrices, 38 c JÍ the subalgebra of infinite matrices with the finite norm, where ||5|| := sup" \J,k \ank\. Let ST C JH be the subalgebra consisting of lower triangular matrices. HausdorfT [7] showed that, if 77 is any HausdorfT matrix with distinct diagonal entries, then the commutant of H in ¿7" consists of HausdorfT matrices.Let T be the subalgebra of Jf consisting of all bounded infinite matrices on c, the space of convergent sequences, A the subalgebra of T composed of triangular matrices.In papers [6] and [7] the author showed that there exist HausdorfT matrices with distinct diagonal entries that have non-Hausdorff matrices in the commutant, and that a necessary condition for the commutant of a HausdorfT matrix in T and A to be the same is that H have distinct diagonal entries.The author [6] and Jakimovski [4] have independently shown that the cornmutant of (C, I), the Cesàro matrix of order 1, in T, is %?, the set of conservative HausdorfT matrices. (A matrix is conservative if it is convergencepreserving over c.) However, the proof only uses the fact that A, (C, 1 ) e 38 . It is well known that, if a HausdorfT matrix has finite norm, then automatically it has all zero column limits, except possibly for the first column, and the first column converges. Since every HausdorfT matrix has row sums equal to po, the Silverman-Toeplitz conditions Tor a matrix to be conservative are satisfied, and the matrix is conservative. Thus the commutants oT (C, I) in A and in 38

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“…Then hnk = (n + a + l)~x, so that Ha is a factorable matrix. (A lower triangular matrix A = (ank) is called factorable if a"¿ = bnck when k < n.) By [7,Theorem 2], any matrix which formally commutes with Ha must be triangular and, hence, by the proof of Theorem 198 in [3], must also be a generalized Hausdorff matrix. Thus, the operators on I2 which commute with Ha are all generalized Hausdorff matrices.…”

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confidence: 99%