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We formulate a general approximation probleminvolving reexive and smoothBanach spaces, and give its explicit solution. Two applications are presented—the rst is to the Bounded Completion Problem involving approximation ofHardy class functions, while the second involves the construction of minimal vec-tors and hyperinvariant subspaces of linear operators, generalizing the Hilbertspace technique of Ansari and Eno. 2000 Mathematics Subject Classication: 41A29, 47A15, 46B20, 46E15.Keywords: Constrained approximation, Smoothness, Invariantsubspaces, Hardyspaces,Extremal problems. 1 Introduction The construction of minimal vectors, corresponding to an operator on a Hilbert spacewas introduced by Ansari and Eno [3], who used it to give simpler proofs of theexistence of invariant subspaces for certain classes of operator, including compact op-erators and normal operators. The minimal vectors are given as the solution of certainconstrained approximation problems, and in [9] a more general such problem was for-mulated and solved, which has other applications, notably in the theories of systemsidentication, signal processing and inverse problems (see [8] for a survey of this area).It is possible to dene minimal vectors in a more general Banach space context,although the associated approximation problem is rather more dicult to resolve. An-droulakis [2] generalized the techniques of Ansari and Eno to produce a new su-cient condition for the existence of hyperinvariant subspaces of certain operators on

Approximation in reflexive Banach spaces and applications to the invariant subspace problem