Vol. 32, No. 2, 1970

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Diagonal submatrices of matrix maps

Alfred Esperanza Tong

Vol. 32 (1970), No. 2, 551–559

The first question answered in this paper is; if A;λ μ is a linear operator between sequence spaces, with a matrix representation (aij), does it follow that the associated diagonal matrix (aijδjj) maps λ into μ? An affirmative answer is given if λ is a normal (or monotone) sequence space and μ is a perfect sequence space.

Morever, if λ,μ are normed sequence spaces, under what conditions will the following inequality hold for all matrix maps (aij) from λ to μ : (αij)(aijδij) (where ∥⋅∥ denotes the operator sup norm)?

We apply our answer to the first problem to give another proof for a theorem of S. Mazur.

Mathematical Subject Classification
Primary: 47.10
Received: 18 February 1969
Published: 1 February 1970
Alfred Esperanza Tong