In this paper we present a family of analysis and synthesis systems of operators with frame-like properties for the range of a bounded operator on a separable Hilbert space. This family of operators is called a Θ–g-frame, where Θ is a bounded operator on a Hilbert space. Θ–g-frames are a generalization of g-frames, which allows to reconstruct elements from the range of Θ. In general, range of Θ is not a closed subspace. We also construct new Θ–g-frames by considering Θ–g-frames for its components. We further study Riesz decompositions for Hilbert spaces, which are a generalization of the notion of Riesz bases. We define the coefficient operators of a Riesz decomposition and we will show that these coefficient operators are continuous projections. We obtain some results about stability of Riesz decompositions under small perturbations.

We investigate the concept of amenability modulo an ideal of Banach algebra, showing that amenability modulo an ideal can be characterized by the existence of virtual and approximate diagonal modulo an ideal. We also study the concept of contractible modulo an ideal of Banach algebra. As a consequence, we prove a version of Selivanov's theorem for a large class of semigroups, includingE-inversiveE-semigroup and eventually inverse semigroups.

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We present a generalization of several fixed and common fixed point theorems on the c-distance in ordered cone metric spaces. In this way, we improve and generalize various results existing in the literature. Наведено узагальнення деяких теорем про нерухому точку та спiльну нерухому точку для c-вiдстанi в упорядкованих конiчних метричних просторах. Таким чином, покращено та узагальнено рiзноманiтнi результати, що наведенi в лiтературi.