Let H be a complex Hilbert space. Denote by B(H)+ the set of all positive bounded linear operators on H. A bijective map φ : B(H)+ → B(H)+ is said to preserve Lebesgue decompositions in both directions if for any quadruple A,B,C,D of positive operators, B = C +D is an A-Lebesgue decomposition of B if and only if φ(B) = φ(C)+φ(D) is a φ(A)-Lebesgue decomposition of φ(B). It is proved that every ...

Let T be a quadratic operator on a complex Hilbert space H. We show that T can be written as a product of two positive contractions if and only if T is of the form aI ⊕ bI ⊕ ( aI P 0 bI ) on H1 ⊕H2 ⊕ (H3 ⊕H3) for some a, b ∈ [0, 1] and strictly positive operator P with ‖P‖ ≤ | √ a− √ b| √ (1− a)(1− b). Also, we give a necessary condition for a bounded linear operator T with operator matrix ( T1...

We propose a discrete time approach for the exponential stability of evolution families on a Hilbert space by proposing a Liapunov-type equation which involves only discrete time arguments. The result of A.M. Lyapunov has come into widespread usage in many topics of mathematics. In particular, it continues to be of great importance in modern treatments of the asymptotic behaviour of the solutio...

Let H be a complex Hilbert space. Denote by B(H)+ the set of all positive bounded linear operators on H. A bijective map φ : B(H)+ → B(H)+ is said to preserve Lebesgue decompositions in both directions if for any quadruple A,B,C,D of positive operators, B = C +D is an A-Lebesgue decomposition of B if and only if φ(B) = φ(C)+φ(D) is a φ(A)-Lebesgue decomposition of φ(B). It is proved that every ...

This paper focuses on the delay-dependent robust stability of linear neutral delay systems. The systems under consideration are described by functional differential equations, with norm bounded time varying nonlinear uncertainties in the “state” and norm bounded time varying quasi-linear uncertainties in the delayed “state” and in the difference operator. The stability analysis is performed via...

In this paper we construct a bounded strictly positive function õ such that the Liouville property fails for the divergence form operator L ≥ r(õ2r). Since in addition ∆õÛõ is bounded, this example also gives a negative answer to a problem of Berestycki, Caffarelli and Nirenberg concerning linear Schrödinger operators.

$K$-frames which are generalization of frames on Hilbert spaces‎, ‎were introduced‎ ‎to study atomic systems with respect to a bounded linear operator‎. ‎In this paper‎, ‎$*$-$K$-frames on Hilbert $C^*$-modules‎, ‎as a generalization of $K$-frames‎, ‎are introduced and some of their properties are obtained‎. ‎Then some relations‎ ‎between $*$-$K$-frames and $*$-atomic systems with respect to a...