Operator convexity in Krein spaces (Q2439671)

A classical notion in operator theory and matrix analysis is the notion of an operator convex function. This is a real valued continuous function \(f\) on a real interval which satisfies \[ f\big((1-\lambda) A + \lambda B\big) \leq (1-\lambda) f(A) + \lambda f(B), \] for all \(\lambda\in[0,1]\) and all selfadjoint operators \(A\) and \(B\) on a Hilbert space whose spectra are contained in the real interval. This notion is generalized to Krein spaces and selfadjoint operators therein. A selfadjoint operator \(A\) in a Krein space \((\mathcal K, [\cdot,\cdot])\) is an operator which coincides with its adjoint \(A^+\), where this adjoint is taken with respect to the (indefinite) Krein space inner product \([\cdot,\cdot]\). Obviously, two problems arise. First, \(f(A)\) is no longer defined for an arbitrary continuous function and a selfadjoint operator \(A\) in a Krein space as, in general, there is no spectral function for selfadjoint operators in Krein spaces. Second, even if \(f(A)\) and \(f(B)\) exist (e.g., if \(f\) is a polynomial) and if one reads the inequality above with respect to the Krein space inner product \([\cdot,\cdot]\), then, due to the indefiniteness of \([\cdot,\cdot]\), this inequality will only hold for very special functions. Therefore a generalization to Krein spaces is done in the following way: Only analytic functions are considered (and then one can use the Riesz-Dunford integral to define \(f(A)\)) and only \(J\)-positive operators are considered (i.e., selfadjoint operators in a Krein space which are positive with respect to the indefinite inner product). To be more precise, let \(\mathcal U\) be an open set symmetric with respect to \(\mathbb R\) with \(\mathcal U \cap \mathbb R \neq \emptyset\). An analytic function \(f\) defined on \(\mathcal U\) which is real on \(\mathcal U \cap \mathbb R\) is called Krein-operator convex, if \[ f\big((1-\lambda) A + \lambda B\big) \leq^J (1-\lambda) f(A) + \lambda f(B), \] for all \(\lambda\in[0,1]\) and all \(J\)-positive operators \(A\) and \(B\) on any Krein space such that the spectra of \(A\), \(B\) and \((1-\lambda) A + \lambda B\) are contained in \(\mathcal U\). Here, \(A\leq^JB\) is understood with respect to the (indefinite) inner product of the Krein space \((\mathcal K, [\cdot, \cdot])\), i.e., \([Ax,x]\leq[Bx,x]\). For such an \(f\) with \(f(0)=0\), a Jensen type inequality is shown, namely, \[ f(C^+AC) \leq^J C^+ f(A)C \] for all \(J\)-positive operators \(A\) and all invertible \(J\)-contractive operators \(C\) (this means that \([Cx,Cx]\leq [x,x]\) for all \(x\in \mathcal K\)) such that the spectra of \(A\), \(C^+AC\) and \(D^+AD\) are contained in \(\mathcal U\), where \(D\) is an injective bounded operator with \(I= CC^+ +DD^+\).