Characterizations of Inner Product Spaces (Operator Theory by Amir

By Amir

Each mathematician operating in Banaeh spaee geometry or Approximation conception is familiar with, from his personal experienee, that the majority "natural" geometrie houses may perhaps faH to carry in a generalnormed spaee until the spaee is an internal produet spaee. To reeall the weIl identified definitions, this suggests IIx eleven = *, the place is an internal (or: scalar) product on E, Le. a functionality from ExE to the underlying (real or eomplex) box gratifying: (i) O for x o. (ii) is linear in x. (iii) = (intherealease, thisisjust =

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Since v(z) ≤ vr (z), u(z) is a harmonic majorant of v(z). 3, U (z) ≥ vr (z) for each r. Consequently, supr vr (0) < ∞, and again u(z) = limr vr (z) is finite and harmonic. Since vr (z) ≤ U (z), we have u(z) ≤ U (z), and so u(z) is the least harmonic majorant. Since by continuity u(z) = limr →1 u(r z), the least harmonic majorant of v(z) can also be written u(z) = lim r →1 Pz (θ)v(r eiθ ) dθ/2π. In particular, if v(z) ≥ 0 and if v(z) has a harmonic majorant, then its least harmonic majorant is the Poisson integral of the weak-star limit of the bounded positive measures v(r eiθ ) dθ/2π .

Hence E is dense in (z 0 , r ). Because E is closed this means (z 0 , r ) ⊂ E, and E is open. Since W was assumed to be connected, we have a contradiction and we conclude that a ≤ 0. Conversely, let z 0 ∈ and let (z 0 , r ) ⊂ . Since v is upper semicontinuous there are continuous functions u n (z) decreasing to v(z) on ∂ (z 0 , r ) as n → ∞. Let Un (z) be the harmonic function on (z 0 , r ) with boundary values u n (z). After a suitable change of scale, Un is obtained from u n by the Poisson integral formula for the unit disc.

Letting n → ∞ now yields 1 2π |B(eiθ )|dθ = 1. so that |B(eiθ )| = 1 almost everywhere. The purpose of the convergence factors −¯z n /|z n | is to make arg bn (z) converge. To remember the convergence factors, note that they are chosen so that bn (0) > 0. 2, the analytic function f (z) has a factorization f (z) = B(z)g(z), z ∈ D, where B(z) is a Blaschke product and where g(z) has no zeros on D, if and only if the subharmonic function log | f (z)| has a harmonic majorant. 3) yn < ∞, 1 + |z n |2 z n = xn + i yn , and the Blaschke product with zeros {z n } is B(z) = z−i z+i m z n =i |z n2 + 1| z − z n .

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