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**Generalized orthogonal decomposition theorem in Banach space and generalized orthogonal complemented subspace.**
*(Chinese.
English summary)*
Zbl 1027.46013

In a Hilbert space \(H\), the Riesz orthogonal decomposition theorem says that any closed linear subspace \(L\subset H\) has a unique orthogonal complement, i.e., a subspace \(M\) of \(H\) such that \(H=L\oplus M\) and \(M\perp L\). In this paper, the authors study the generalization of the Riesz theorem to Banach spaces. We call vectors \(x,y\) in a normed space \(X\) orthogonal and denote this by \(x\perp y\), if \(d_{\langle y\rangle }( x) =\|x\|\), where \(\langle y\rangle \) is the linear span of \(y\) and \(d_{S}( x) =\inf_{s\in S}\|x-s\|\) is the shortest distance from \(x\) to a set \(S\subset X\), and we also call two subsets \(A,B\subset X\) orthogonal and denote this by \( A\perp B\), if \(d_{B}( x) =\|x\|\) for all \(x\in A\). Note that in a general Banach space \(X\), the orthogonal condition \(A\perp B\) is not symmetric in \(A,B\), but in a Hilbert space it does coincide with the traditional orthogonal condition. A (closed) linear subspace \(L\subset X\) is called orthogonally complementable if there is a (closed) linear subspace \(M\) with \(X=L\oplus M\) and \(M\perp L\).

The authors obtain results on the orthogonal decomposition in Banach algebras under suitable conditions. In particular, it is shown that a closed subspace \(L\) of a strictly convex space \(X\) is orthogonally complementable if and only if \(L\) is a Chebyshev subspace of \(X\) and \(F_{X}^{-1}( L^{\perp }) =\{ x\in X\mid F_{X}( x) \cap L^{\perp }\neq \emptyset \} \) is an additive set, where \(F_{X}( x) =\{ x^{\ast }\in X^{\ast }\mid \langle x^{\ast },x\rangle =\|x\|^{2}\} \) and \(L^{\perp }=\{ x^{\ast }\in X^{\ast }\mid \langle x^{\ast },x\rangle =0 \text{for all }x\in L\} \).

The authors obtain results on the orthogonal decomposition in Banach algebras under suitable conditions. In particular, it is shown that a closed subspace \(L\) of a strictly convex space \(X\) is orthogonally complementable if and only if \(L\) is a Chebyshev subspace of \(X\) and \(F_{X}^{-1}( L^{\perp }) =\{ x\in X\mid F_{X}( x) \cap L^{\perp }\neq \emptyset \} \) is an additive set, where \(F_{X}( x) =\{ x^{\ast }\in X^{\ast }\mid \langle x^{\ast },x\rangle =\|x\|^{2}\} \) and \(L^{\perp }=\{ x^{\ast }\in X^{\ast }\mid \langle x^{\ast },x\rangle =0 \text{for all }x\in L\} \).

Reviewer: Albert Sheu (Lawrence)