Obviously, the nonexpansive mapping class is a proper subclass of the strict pseudo-contraction class and the Lipschitzian operator class is a proper subclass of the boundedly Lipschitzian operator class, respectively.
In 2004, Berinde [55] introduced an almost contraction, a new class of contractive type mappings which exhibits totally different features more than the one of the particular results incorporated [1, 16, 39, 50], i.e., an almost contraction generally does not have a unique fixed point; see Example 1 in [55].
Since strong stability coincides with proper contractiveness for non-negative operators [10], (D_{i_{0}}) is a proper contraction (of the class (C_{00}) of contractions).
Recently Duggal et al. [11] showed that if T is a class contraction, then either T has a non-trivial invariant subspace or T is a proper contraction and the nonnegative operator D = | T 2 | − | T | 2 is strongly stable.
We answer a question from K. R. Davidson and V. I. Paulsen (1997, J. Reine Angew Math. 487, 153 170) about the similarity to a contraction of a class of CAR-valued Foguel–Hankel operator.
They also discussed quasi-contraction, almost contraction and the class of mappings that satisfy condition (B) in detail.
Our main theorem establishes a relation between semigroups of quasi-sectorial contractions and a class of m-sectorial generators.
The study is carried out in the setting of the Hardy space over the bidisk H2(D2), on which every C⋅0-class contraction has a representation.
In the present paper, a simplification is achieved by extending the given operator (a positive L1-contraction) to the class of all (i.e., not necessarily measurable) functions on the underlying measure space.
(ii) Let T be a contraction of ∗-class A operators.
