We show that a compact operator A is a multiple of a positive semi-definite operator if and only if (A B) W (A) W (B), for all (rank one) operators B. An example of a normal operator is given to show that the equivalence conditions may fail in general. We then obtain conditions to identify other classes of operators A so that equivalence conditions hold.

A convex subset B of a real locally convex space B is said to have the separation property if it can be separated from every closed convex subset B of B , which is disjoint from B , by a closed hyperplane. The strong separation theorem says that if B is weakly compact, then it has the separation property. In this paper, we present two versions of the converse and discuss an application of them. For example, we prove that a normed space is reflexive if and only if its closed unit ball has the separation property. Results in this paper can be considered as supplements of the famous theorem of James.

In this paper, we try to attack a conjecture of Araujo and Jarosz that every bijective linear map between C-algebras, with both and its inverse 1 preserving zero products, arises from an algebra isomorphism followed by a central multiplier. We show it is true for CCR C-algebras with Hausdorff spectrum, and in general, some special C-algebras associated to continuous fields of C-algebras.

We introduce the Jordan product associated with the second-order cone K into the real Hilbert space H, and then define a one-parametric class of complementarity functions t on H H with the parameter t[0, 2). We show that the squared norm of t with t(0, 2) is a continuously F (rchet)-differentiable merit function. By this, the second-order cone complementarity problem (SOCCP) in H can be converted into an unconstrained smooth minimization problem involving this class of merit functions, and furthermore, under the monotonicity assumption, every stationary point of this minimization problem is shown to be a solution of the SOCCP.

Recently, Kawasaki and Takahashi (J. Nonlinear Convex Anal. 14:71-87, 2013) defined a broad class of nonlinear mappings, called widely more generalized hybrid, in a Hilbert space which contains generalized hybrid mappings (Kocourek <i>et al.</i> in Taiwan. J.Math. 14:2497-2511, 2010) and strict pseudo-contractive mappings (Browder and Petryshyn in J. Math. Anal. Appl. 20:197-228, 1967). They proved fixed point theorems for such mappings. In this paper, we prove fixed point theorems for widely more generalized hybrid non-self mappings in a Hilbert space by using the idea of Hojo <i>et al.</i> (Fixed Point Theory 12:113-126, 2011) and Kawasaki and Takahashi fixed point theorems (J.Nonlinear Convex Anal. 14:71-87, 2013). Using these fixed point theorems for non-self mappings, we proved the Browder and Petryshyn fixed point theorem (J.Math. Anal. Appl. 20:197-228, 1967) for strict pseudo-contractive