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Adjoint subspaces in Banach spaces, with applications to ordinary differential subspaces

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Summary

Given two subspaces A0 ⊂ A1 ⊂ W=X ⊕ Y, where X, Y are Banach spaces, we show how to characterize, in terms of generalized boundary conditions, those adjoint pairs A, A* satisfying A0 ⊂ A ⊂ A1, A 1 * ⊂ A∗ ⊂ A 0 * ⊂ W+=Y* ⊕ X*, where X*, Y* are the conjugate spaces of X, Y, respectively. The characterizations of selfadjoint (normal) subspace extensions of symmetric (formally normal) subspaces appear as special cases when Y=X*. These results are then applied to ordinary differential subspaces in W=Lq(ι) ⊕ Lr(ι), 1≦q, r≦∞, where τ is a real interval, and in W=C(\(\bar \iota \)) ⊕ C(\(\bar \iota \)), where\(\bar \iota \) is a compact interval.

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Entrata in Redazione il 21 febbraio 1977.

The work of EarlA. Coddington was supported in part by the National Science Foundation under NSF Grant No. MCS-76-05855.

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Coddington, E.A., Dijksma, A. Adjoint subspaces in Banach spaces, with applications to ordinary differential subspaces. Annali di Matematica 118, 1 (1978). https://doi.org/10.1007/BF02415124

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Keywords

  • Boundary Condition
  • Banach Space
  • Generalize Boundary
  • Compact Interval
  • Generalize Boundary Condition