Vol. 17, No. 3, 1966

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Contractive projections in Lp spaces

Tsuyoshi Andô

Vol. 17 (1966), No. 3, 391–405

Let ,A,m) be a finite measure space and Lp(1 < p < ) the Lebesgue space of all complex valued measurable functions whose absolute p-th powers are integrable. Given a closed linear subspace of Lp, the operator which assigns to f the function in the subspace with minimum distance from it is continuous, idempotent, but not linear in general except the case p = 2 when the operator is just an orthogonal projection. A problem is to determine when such an operator Q is linear. It is linear if and only if P = I Q is a contractive projection, i.e., a linear idempotent operator with P1, so that the problem takes an equivalent form to give complete description of contractive projections in Lp. In this paper the problem will be settled in the latter form, not only for 1 < p < but also for 0 < p 1.

Mathematical Subject Classification
Primary: 47.25
Received: 18 January 1965
Published: 1 June 1966
Tsuyoshi Andô