For the usual norm on spaces C(K) and C b (Ω) of all continuous functions on a compact Hausdorff space K (all bounded continuous functions on a locally compact Hausdorff space Ω), the following equalities are proved: lim t→0+ ||f + tg|| C(K) − ||f || C(K) t = max x∈{z | |f (z)|=||f ||} Re(e −i arg f (x) g(x)). and lim t→0+ ||f + tg|| C b (Ω) − ||f || C b (Ω) t = inf δ>0 sup x∈{z | |f (z)|≥||f |...

We prove the long standing conjecture in the theory of two-point boundary value problems that completeness and Dunford’s spectrality imply Birkhoff regularity. In addition we establish the even order part of S.G.Krein’s conjecture that dissipative differential operators are Birkhoff-regular and give sharp estimate of the norms of spectral projectors in the odd case. Considerations are based on ...

A new constantWD(X) is introduced into any real 2n-dimensional symmetric normed space X . By virtue of this constant, an upper bound of the geometric constant D(X), which is used to measure the difference between Birkhoff orthogonality and isosceles orthogonality, is obtained and further extended to an arbitrarym-dimensional symmetric normed linear space (m≥ 2). As an application, the result is...

A Banach space theoretical characterization of abelian C ∗ C^* - ...