In the present paper, monotone relations and maximal monotone relations from an Hadamard space to its linear dual space are investigated. Fitzpatrick transform of monotone relations in Hadamard spaces is introduced. It is shown that Fitzpatrick transform of a special class of monotone relations is proper, convex and lower semi-continuous. Finally, a representation result for monotone relations is given.

Approximately dual frames as a generalization of duality notion in Hilbert spaces have applications in Gabor systems, wavelets, coorbit theory, and sensor modeling. In recent years, the computing of the associated deviations of the canonical and alternate dual frames from the original ones has been considered by functional analysts. In this paper, we consider the quantitative measurement of the associated deviations of the alternate and approximately dual frames from the original ones. More precisely, it is proved that if the sequence Ψ is sufficiently close to the frame Φ, then for given approximately dual frame Φ ad of Φ, the approximately dual frame Ψ ad of Ψ can be found which is close to Φ ad , and particularly, we estimate their deviation in terms of the distance of the operators  1 ∶= T Φ U Φ ad and  2 ∶= T Ψ U Ψ ad and their approximation rates, where T X and U X denote the synthesis and analysis operators of the frame X, respectively. It is worth mentioning that some of our perturbation conditions are quite different from those used in the previous literatures on this topic.

Let A be a Banach algebra, and X a Banach A-bimodule. A bounded linear mapping D : A → X is approximately semi-inner derivation if there eixist nets (ξ α ) α and (µ α ) α in X such that, for each a ∈ A, D(a) = lim α (a.ξ α − µ α .a). A is called approximately semi-amenable if for every Banach A-bimodule X , every D ∈ Z 1 (A, X * ) is approximtely semi-inner. There are some Banach algebras which are approximately semi-amenable, but not approximately amenable. In this manuscript, we investigate some properties of approximate semi-amenability of Banach algebras. Also in Theorem 2.7 we prove the approximate semi-amenability of Segal algebras on a locally compact group G.

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We investigate -approximate contractibility and -approximate amenability of Banach algebras, which are extensions of usual notions of contractibility and amenability, respectively, where is a dense range or an idempotent bounded endomorphism of the corresponding Banach algebra.