An n-Dimensional Hahn-Banach Extension Theorem and Minimal Projections
Tóm tắt
Let T~=∑i=1
n ũi⊗rvi:V → V=[v1,. . . .,vn]⊂ X, where ũi∈ V* and X is a Banach space. Let T= ∑i=1
nui⊗vi: X→ V be an extension of T~ to all of X (i.e., ui∈ X*) such that T has minimal (operator) norm. (E.g., if T~=I, T is a minimal projection from X onto V.) Then it is necessary and sufficient that u:=u_1,. . . ,un is given by (v:=v1,. . . ,vn) extv(u)∈ Vn,where the notion of a v-extremal (“extv”) of u is properly defined. The condition above leads in many important cases to a simple geometric interpretation of minimal projections. Furthermore, by applying this formula to the case X=Lp, we obtain a linear n-dimensional analog of the Hölder equality condition (M is given by extv(u)=Mv) 1/p u′ · Mv = 1/q u · Mv′, wherever v is differentiable. We point out several applications, including the determination of the absolute projection constant of ℓ
Tài liệu tham khảo
B. L. Chalmers, The (*)-equation and the form of the minimal projection operator, in Approximation Theory IV (C. K. Chui, et al., eds.) Academic Press, New York, 1983, pp. 393–399.
B. L. Chalmers, The n-dimensional Hölder inequality, submitted.
B. L. Chalmers, The absolute projection constant of ℓp n, in preparation.
B. L. Chalmers and C. Franchetti, The determination of a minimal L p projection onto the lines, in preparation.
B. L. Chalmers, C. Franchetti, and M. Giaquinta, On the self-length of two-dimensional Banach spaces, Bull. Austr. Mat. Soc., 53, 101–107 (1996).
B. L. Chalmers and G. Lewicki, Minimal projections onto some subspaces of ℓp (n) 1 , Functiones ed Approximatio, 26, 85–92 (1998).
B. L. Chalmers and G. Lewicki, Two-dimensional real symmetric spaces with maximal projection constants, Ann Polonici. Math. 73, 119–134 (2000).
B. L. Chalmers and G. Lewicki, Symmetric spaces with maximal projection constants, J. Funct. Anal., to appear.
B. L. Chalmers and G. Lewicki, Minimal projections onto symmetric spaces with large projection constants, Studia Math. 134, 119–133 (1999).
B. L. Chalmers, D. Leviatan, and M. P. Prophet, Optimal interpolating spaces preserving shape, J. Approx. Thy. 98, 354–373 (1999).
B. L. Chalmers and F. T. Metcalf, A simpleformula showing L 1 is a maximal overspace for two-dimensional real spaces, Ann. Polonici Math. 56, 303–309 (1992).
B. L. Chalmers and F. T. Metcalf, The minimal projection from L 1 onto π n , in Stochastic Processes and Functional Analysis (Goldstein, Gretsky, Uhl, eds.) Marcel Dekker, Inc., New York, 1996, pp. 61–69.
B. L. Chalmers and F. T. Metcalf, The determination of minimal projections and extensions in L 1, Trans. Amer. Math. Soc. 329, 289–305 (1992).
B. L. Chalmers and F. T. Metcalf, A characterizationand equations for minimal projections and extensions, J. Operator Theory, 32, 31–46 (1994).
B. L. Chalmers and F. T. Metcalf, Determination of a minimal projection from C[-1, 1] onto the quadratics, Num. Func. Anal. and Optim., 11, 1–10 (1990).
B. L. Chalmers and F. T. Metcalf, Determination of a minimal projection from C[-1, 1] onto πn, in preparation.
B. L. Chalmers and F. T. Metcalf, Construction of minimal projections, in Approximation Theory VIII (C. K. Chui and L. Schumaker, eds.) Academic Press, New York, 1995, pp. 119–127.
B. L. Chalmers and B. Shekhtman, A two-dimensional Hahn-Banach theorem, Proc. A.M.S. 129, 719–724 (2001).
B. Chalmers and B. Shekhtman, Extension constants of unconditional two-dimensional operators, Lin. Alg. and Appl. 240, 173–182 (1996).
B. L. Chalmers and B. Shekhtman, Actions that characterize ℓn ∞, Lin. Alg. and Appl. 270, 155–169 (1998).
L. E. Dor, Potentials and isometric embeddings in L 1, Israel J. Math. 24, 260–268 (1976).
C. Franchetti and E. W. Cheney, Minimal projections in L1-space, Duke Math. J. 43, 501–510 (1976).
J. Lindenstrauss, On theextension of operators with a finite-dimensional range, Illinois J. Math. 8, 488–499 (1964).
D. Yost, L1 contains every two-dimensional normed space, Ann. Polonici Math. 49, 17–19 (1988).