Operators of 𝑃-variation and the evolution representation problem

Freedman, M. A.

doi:10.1090/S0002-9947-1983-0704604-2

Operators of $P$-variation and the evolution representation problem
HTML articles powered by AMS MathViewer

by M. A. Freedman PDF

Trans. Amer. Math. Soc. 279 (1983), 95-112 Request permission

Abstract:

In contrast to a continuous linear semigroup, a continuous linear evolution $U( \cdot )$ may be nondifferentiable or of unbounded variation. In order to study these evolutions we introduce a class of operator-valued functions $A( \cdot )$ which satisfy a generalized bounded variation condition and represent $U$ as the product integral $U = \prod [I + dA]$.

References

J. C. Burkill and F. W. Gehring, A scale of integrals from Lebesgue’s to Denjoy’s, Quart. J. Math. Oxford Ser. (2) 4 (1953), 210–220. MR 57307, DOI 10.1093/qmath/4.1.210
Michel Bruneau, Variation totale d’une fonction, Lecture Notes in Mathematics, Vol. 413, Springer-Verlag, Berlin-New York, 1974 (French). MR 0393380
M. A. Freedman, A faithful Hille-Yosida theorem for finite-dimensional evolutions, Trans. Amer. Math. Soc. 265 (1981), no. 2, 563–573. MR 610966, DOI 10.1090/S0002-9947-1981-0610966-5
M. A. Freedman, Necessary and sufficient conditions for discontinuous evolutions with applications to Stieltjes integral equations, J. Integral Equations 5 (1983), no. 3, 237–270. MR 702433
F. W. Gehring, A study of $\alpha$-variation. I, Trans. Amer. Math. Soc. 76 (1954), 420–443. MR 64859, DOI 10.1090/S0002-9947-1954-0064859-9
J. V. Herod and R. W. McKelvey, A Hille-Yosida theory for evolutions, Israel J. Math. 36 (1980), no. 1, 13–40. MR 589655, DOI 10.1007/BF02761228
E. R. Love, A generalization of absolute continuity, J. London Math. Soc. 26 (1951), 1–13. MR 39791, DOI 10.1112/jlms/s1-26.1.1
J. S. MacNerney, Integral equations and semigroups, Illinois J. Math. 7 (1963), 148–173. MR 144179
J. S. MacNerney, A linear initial-value problem, Bull. Amer. Math. Soc. 69 (1963), 314–329. MR 146613, DOI 10.1090/S0002-9904-1963-10905-2
D. S. Nathan, One-parameter groups of transformations in abstract vector spaces, Duke Math. J. 1 (1935), no. 4, 518–526. MR 1545897, DOI 10.1215/S0012-7094-35-00139-9

The quadratic variation of a function and its Fourier coefficients

3

L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936), no. 1, 251–282. MR 1555421, DOI 10.1007/BF02401743