We consider function spaces by analogy to Sobolev spaces and spaces of smooth functions on a finite-dimensional Euclidean space. For a real separable Hilbert space E we introduce the Hilbert space 
of complex-valued functions on E that are square integrable with respect to some measure λ invariant under shifts and orthogonal transformations of E. For one-parameter semigroups of selfadjoint contractions in H we obtain the strong continuity criterion and study properties of their generators. For counterparts of Sobolev spaces and spaces of smooth functions we find necessary and sufficient embedding conditions and obtain the existence conditions for traces of functions in the Sobolev spaces on hyperplanes of codimension 1 in the space E. Bibliography: 13 titles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. Zh. Sakbaev, “Averaging of random walks and shift-invariant measures on a Hilbert space,” Theor. Math. Phys. 191, No. 3, 886–909 (2017).
Kuo, Hui-Hsiung Gaussian Measures in Banach Spaces, Springer, Berlin etc. (1975).
A. M. Vershik, “Does there exist a Lebesgue measure in the infinite-dimensional space?” Proc. Steklov Inst. Math. 259, 248–272 (2007).
R. Baker, “Lebesgue measure on R∞,” Proc. Am. Math. Soc. 113, No. 4, 1023–1029 (1991).
V. Zh. Sakbaev, “Random walks and measures on Hilbert space that are invariant with respect to shifts and rotations,” J. Math. Sci., New York 241, No. 4, 469–500 (2019).
V. I. Bogachev, Measure Theory. I, Springer, Berlin (2007).
N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, John Wiley& Sons, New York etc. (1988).
V. I. Bogachev, Gaussian Measures Am. Math. Soc., Providence, RI (1998).
V. Zh. Sakbaev, “Semigroup of transformations of the space of square integrable functions with respect to translation invariant measures on a Banach space” [in Russian], Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 151, 73–90 (2018).
M. G. Sonis, “ On measurable subspaces of the space of all sequences with a Gaussian measure” [in Russian], Uspekhi Mat. Nauk 21, No. 5, 277–279 (1966).
V. V. Zhikov, “Weighted Sobolev spaces,” Sb. Math. 189, No. 8, 1139–1170 (1997).
T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin (1995).
J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer, Berlin etc. (1973).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Problemy Matematicheskogo Analiza 103, 2020, pp. 43-62.
Rights and permissions
About this article
Cite this article
Busovikov, V.M., Sakbaev, V.Z. Shift-Invariant Measures on Hilbert and Related Function Spaces. J Math Sci 249, 864–884 (2020). https://doi.org/10.1007/s10958-020-04980-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-020-04980-1