Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A class of functional equations and Mielnik probability spaces

Authors: S. J. Guccione and Č. V. Stanojević
Journal: Proc. Amer. Math. Soc. 59 (1976), 317-320
MSC: Primary 46C10
MathSciNet review: 0454605
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ S$ be the unit sphere of a normed real linear space $ N$ and let $ (S,p)$ be a Mielnik space of dimension two. For $ p(x,y) = f(\vert\vert x + y\vert\vert),\;x,\;y \in S$, where $ f$ is a continuous, strictly increasing function from $ [0,2]$ onto $ [0,1]$, it has been shown that $ (S,p)$ being two dimensional is equivalent to $ N$ being an inner product space. In some polarization problems modeled on the unit sphere of an inner product space, the transition probability $ p(x,y)$ may not be as well behaved as $ p(x,y) = f(\vert\vert x + y\vert\vert)$. In order to provide a more suitable setting, we have constructed wide classes of two-dimensional transitional probability spaces $ (S,p)$, all having the same set of bases $ \mathcal{B}$, with $ p = \phi \circ f$ where $ \phi $ is a solution of a certain functional equation. In particular, for $ p(x,y) = \vert\vert x + y\vert\vert{\raise0.5ex\hbox{$\scriptstyle 2$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}}$, we answer a question due to B. Mielnik.

References [Enhancements On Off] (What's this?)

  • [1] Č. V. Stanojević, Mielnik probability spaces and characterization of inner product spaces, Trans. Amer. Math. Soc. 183 (1973), 441-448. MR 48 #6904. MR 0328562 (48:6904)
  • [2] B. Mielnik, Geometry of quantum states, Comm. Math. Phys. 9 (1968), 55-80. MR 37 #7156. MR 0231603 (37:7156)
  • [3] P. L. Ganguli, A note on the integral of some classes of functions, Bull. Calcutta Math. Soc. 57 (1965), 63-68. MR 35 #1733. MR 0210848 (35:1733)
  • [4] A. R. Blass and Č. V. Stanojević, Partial Mielnik spaces and characterization of uniformly convex spaces, Proc. Amer. Math. Soc. 55 (1976), 75-82. MR 0394121 (52:14926)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46C10

Retrieve articles in all journals with MSC: 46C10

Additional Information

Keywords: Mielnik probability spaces, functional equation
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society