Recently, the 4-hand expanded endoscopic approach through the nose to address tumors arising endocranially, with or without skull base involvement, has become extended, proving the nose to be a perfect access for tumors localized centrally and thus avoiding external approaches associated with higher morbidity [ 2].
In Section 3, we extend inequalities proved in Section 2 from the scalars setting to a Hilbert space operator setting.
Remark 6 Our theorems in this paper are important generalizations of several important recent results in the following sense: (i) Our theorems extend results proved for multi-valued nonexpansive mappings in real Hilbert spaces (see, e.g., [15 17, 20, 21]) to a much larger class of multi-valued strictly pseudo-contractive mappings and in a much larger class of q-uniformly smooth real Banach spaces.
As an application we determine the essential commutant of the set of all analytic Toeplitz operators on H2 and thus extend results proved by Davidson (1977) [6] for the unit disc and Ding and Sun (1997) [11] for the unit ball.
In this paper, we extend and prove Ky Fan's Theorem for discontinuous increasing maps f in a Banach lattice X when f has no compact conditions.
Particularly, they extended and proved the Marotto theorem to general Banach spaces [14], established some chaotification theorems for discrete dynamical systems in (infinite-dimensional) Banach spaces [12], and showed the controlled systems are chaotic in the sense of both Devaney and Li-Yorke.
We extend and prove Abrahamseʼs theorem to the case of matrix-valued symbols; that is, we show that every subnormal block Toeplitz operator with bounded type symbol (i.e., a quotient of two bounded analytic functions), whose analytic and co-analytic parts have the "left coprime factorization", is normal or analytic.
All we are doing is extending this proven way of working.
The Etom-1 well in the northern part of the Lokichar basin discovered 10 meters of net oil pay extending the proven oil basin northwards.
The development extends a proven numerical framework for integrating the soundproof equations.
