Geodesic
Approximation of Operator Semigroups Using Linear-Fractional Operator Functions and Weighted Averages
Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 2, pp. 47-63.

Voir la notice de l'article provenant de la source Math-Net.Ru

An analytic semigroup of operators on a Banach space is approximated by a sequence of positive integer powers of a linear-fractional operator function. It is proved that the order of the approximation error in the domain of the generating operator equals $O(n^{-2}\ln(n))$. For a self-adjoint positive definite operator $A$ decomposed into a sum of self-adjoint positive definite operators, an approximation of the semigroup {$\exp(-tA)$} ($t\geq0$) by weighted averages is also considered. It is proved that the order of the approximation error in the operator norm equals $O(n^{-1/2}\ln(n))$.
Keywords: approximation of semigroup, Trotter–Chernoff formula, analytic semigroup. 
@article{FAA_2022_56_2_a4,
     author = {D. L. Rogava},
     title = {Approximation of {Operator} {Semigroups} {Using} {Linear-Fractional} {Operator} {Functions} and {Weighted} {Averages}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {47--63},
     publisher = {mathdoc},
     volume = {56},
     number = {2},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2022_56_2_a4/}
}
TY  - JOUR
AU  - D. L. Rogava
TI  - Approximation of Operator Semigroups Using Linear-Fractional Operator Functions and Weighted Averages
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2022
SP  - 47
EP  - 63
VL  - 56
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2022_56_2_a4/
LA  - ru
ID  - FAA_2022_56_2_a4
ER  - 
%0 Journal Article
%A D. L. Rogava
%T Approximation of Operator Semigroups Using Linear-Fractional Operator Functions and Weighted Averages
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2022
%P 47-63
%V 56
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2022_56_2_a4/
%G ru
%F FAA_2022_56_2_a4
D. L. Rogava. Approximation of Operator Semigroups Using Linear-Fractional Operator Functions and Weighted Averages. Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 2, pp. 47-63. http://geodesic.mathdoc.fr/item/FAA_2022_56_2_a4/

[1] H. Trotter, “On the product of semi-group of operators”, Proc. Amer. Mat. Soc., 10 (1959), 545–551 | DOI | MR | Zbl

[2] P. R. Chernoff, “Note on product formulas for operator semigroups”, J. Funct. Anal., 2 (1968), 238–242 | DOI | MR | Zbl

[3] P. R. Chernoff, “Semigroup product formulas and addition of unbounded operators”, Bull. Amer. Mat. Soc., 76 (1970), 395–398 | DOI | MR | Zbl

[4] Z. G. Gegechkori, J. L. Rogava, M. A. Tsiklauri, “High degree precision decomposition method for the evolution problem with an operator under a split form”, M2AN Math. Model. Numer. Anal., 36:4 (2002), 693–704 | DOI | MR | Zbl

[5] Z. G. Gegechkori, J. L. Rogava, M. A. Tsiklauri, “The fourth order accuracy decomposition scheme for an evolution problem”, M2AN Math. Model. Numer. Anal., 38:4 (2004), 707–722 | DOI | MR | Zbl

[6] Dzh. L. Rogava, “O pogreshnosti formul tipa Trottera v sluchae samosopryazhennykh operatorov”, Funkts. analiz i ego pril., 27:3 (1993), 84–86 | MR | Zbl

[7] T. Ichinose, H. Tamura, “The norm convergence of the Trotter–Kato product formula with error bound”, Comm. Math. Phys., 217:3 (2001), 489–502 | DOI | MR | Zbl

[8] T. Ichinose, Hideo Tamura, Hiroshi Tamura, V. A. Zagrebnov, “Note on the paper “The norm convergence of the Trotter–Kato product formula with error bound” by T. Ichinose and H. Tamura [Comm. Math. Phys., 217:3 (2001), 489–502]”, Comm. Math. Phys., 221:3 (2001), 499–510 | DOI | MR | Zbl

[9] H. Neidhardt, V. A. Zagrebnov, “On error estimates for the Trotter–Kato product formula”, Lett. Math. Phys., 44:3 (1998), 169–186 | DOI | MR | Zbl

[10] T. Ichinose, H. Neidhardt, V. A. Zagrebnov, “Trotter–Kato product formula and fractional powers of self-adjoint generators”, J. Funct. Anal., 207:1 (2004), 33–57 | DOI | MR | Zbl

[11] V. Cachia, V. A. Zagrebnov, “Operator-norm approximation of semigroups by quasi-sectorial contractions”, J. Funct. Anal., 180:1 (2001), 176–194 | DOI | MR | Zbl

[12] A. Gomilko, Yu. Tomilov, “On convergence rates in approximation theory for operator semigroups”, J. Funct. Anal., 266:5 (2014), 3040–3082 | DOI | MR | Zbl

[13] Yu. Arlinskii, V. Zagrebnov, “Numerical range and quasi-sectorial contractions”, J. Math. Anal. Appl., 366:1 (2010), 33–43 | DOI | MR | Zbl

[14] V. Bentkus, V. Paulauskas, “Optimal error estimates in operator-norm approximations of semigroups”, Lett. Math. Phys., 68:3 (2004), 131–138 | DOI | MR | Zbl

[15] E. Khille, R. Fillips, Funktsionalnyi analiz i polugruppy, IL, M., 1962 | MR

[16] T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972

[17] N. Danford, Dzh. Shvarts, Lineinye operatory, Ch. I, IL, M., 1962

[18] L. V. Kantorovich, G. P. Akilov, Funktsionalnyi analiz, Nauka, M., 1977 | MR