Denseness for norm attaining operator-valued functions
Abstract In this note we offer a short, constructive proof for Hilbert spaces of Lindenstrauss' famous result on the denseness of norm attaining operators. Specifically, we show given any A∈ L ( H ) there is a sequence of rank-1 operators K n such that A+K n is norm attaining for each n and K n converges in norm to zero. We then apply our construction to establish denseness results for norm attaining operator-valued functions.
Denseness of norm attaining mappings
The Bishop-Phelps Theorem states that the set of (bounded and linear) functionals on a
Banach space that attain their norms is dense in the dual. In the complex case, Lomonosov proved that
there may be a closed, convex and bounded subset C of a Banach space such that the set of functionals
whose maximum modulus is attained on C is not dense in the dual. This paper contains a survey of
versions for operators, multilinear forms and polynomials of the Bishop-Phelps Theorem. Lindenstrauss
provided examples of Banach spaces X and Y such that the set of norm attaining operators from X to Y
is not dense. He also gave isometric conditions on X for which the set of norm attaining operators from
X to Y are dense in the space of all operators between these Banach spaces. If the above conclusion
holds for every Y , X is said to have the property A. Also, there are known sufficient conditions on the
range space Y in order to have the same denseness conditions for every X. In such a case, Y has the
property B. Bourgain proved that every space satisfying the Radon-Nikod?ym property has the property
A. For classical Banach spaces, it is known that C[0; 1], `p (1 < p < 1) and any infinite-dimensional
L1(¹) do not satisfy the property B (results due to Schachermayer, Gowers and Acosta, respectively).
When both X and Y are either C(K) or L1(¹), there are positive results due to Johnson and Wolfe, and
Iwanik, respectively. Finet and Pay?a proved that there is also positive result for X = L1 and Y = L1.
For multilinear mappings, Aron, Finet andWerner initiated the research and gave sufficient conditions
on a Banach space X in order to satisfy the denseness of the set of norm attaining N-linear mappings in
the set of all the N-linear mappings (Radon-Nikod?ym property, for instance). Choi showed that the space
L1[0; 1] does not satisfy the denseness of the set of norm attaining bilinear forms. Alaminos, Choi, Kim
and Pay?a proved that for any scattered compact space K, the set of norm attaining N-linear forms on
C(K) is dense in the space of all N-linear forms, and for the bilinear case no restriction on the compact
is needed. Acosta, Garc?ýa and Maestre proved that the set of N-linear forms whose Arens extensions
to the bidual attains the norm is dense in the space of all the N-linear forms on a product of N Banach
spaces. For polynomials and for holomorphic mappings, there are some results along the same line, but
more open problems than for the multilinear case.
Inhomogeneous Boundary Value Problems for Compressible Navier-Stokes Equations: Well-Posedness and Sensitivity Analysis
In this paper compressible, stationary Navier–Stokes equations are considered. A framework for analysis of such equations is established. In particular, the well-posedness for inhomogeneous boundary value problems of elliptic-hyperbolic type is shown. Analysis is performed for small perturbations of the so-called approximate solutions that take form (1.12). The approximate solutions are determined from Stokes problem (1.11). The small perturbations are given by solutions to (1.13). The uniqueness of solutions for problem (1.13) is proved, and, in addition, the differentiability of solutions with respect to the coefficients of differential operators is shown. The results on the well-posedness of nonlinear problems are interesting on their own and are used to obtain the shape differentiability of the drag functional for incompressible Navier–Stokes equations. The shape gradient of the drag functional is derived in the classical and useful for computations form; an appropriate adjoint state is introduced to ...