Geodesic
Estimates for the Involution of Decomposable Elements of a Complex Banach Algebra
Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 4, pp. 14-31.

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

An element $a$ of a complex Banach algebra with unit ${1\mspace{-4.85mu}{\mathrm I}}$ and with standard conditions on the norm ($\|ab\|\le\|a\|\cdot\|b\|$ and $\|{1\mspace{-4.85mu}{\mathrm I}}\|=1$) is said to be Hermitian if $\|e^{ita}\|=1$ for any real number $t$. An element is said to be decomposable if it admits a representation of the form $a+ib$ in which $a$ and $b$ are Hermitian. The decomposable elements form a Banach Lie algebra (with respect to the commutator). The Hermitian components are determined uniquely, and hence this Lie algebra has the natural involution $a+ib=x\to x^{*}=a-ib $. One can readily see that $\|x^{*}\|\le2\|x\|$. Among other things, we prove that $\|x^{*}\|\le\gamma \|x\|$, where $\gamma 2$. In fact, the situation is treated in more detail: the original problem is included in a continuous family parametrized by the numerical radius of the element. Finding the exact value of the constant $\gamma $ is reduced to a variational problem in the theory of entire functions of exponential type. Approximately, $\gamma$ is equal to $1.92\pm 0.04$.
Keywords: complex Banach algebra, involution, entire function, variational problem.
Mots-clés : decomposable element 
@article{FAA_2005_39_4_a1,
     author = {E. A. Gorin},
     title = {Estimates for the {Involution} of {Decomposable} {Elements} of a {Complex} {Banach} {Algebra}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {14--31},
     publisher = {mathdoc},
     volume = {39},
     number = {4},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2005_39_4_a1/}
}
TY  - JOUR
AU  - E. A. Gorin
TI  - Estimates for the Involution of Decomposable Elements of a Complex Banach Algebra
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2005
SP  - 14
EP  - 31
VL  - 39
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2005_39_4_a1/
LA  - ru
ID  - FAA_2005_39_4_a1
ER  - 
%0 Journal Article
%A E. A. Gorin
%T Estimates for the Involution of Decomposable Elements of a Complex Banach Algebra
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2005
%P 14-31
%V 39
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2005_39_4_a1/
%G ru
%F FAA_2005_39_4_a1
E. A. Gorin. Estimates for the Involution of Decomposable Elements of a Complex Banach Algebra. Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 4, pp. 14-31. http://geodesic.mathdoc.fr/item/FAA_2005_39_4_a1/

[1] Bonsall F. F., Duncan J., Complete Normed Algebras, Springer-Verlag, New York–Heidelberg–Berlin, 1973 | MR | Zbl

[2] Gorin E. A., “Neravenstva Bernshteina s tochki zreniya teorii operatorov”, Vestnik Kharkov. univ., ser. prikl. mat. i mekh., 205:45 (1980), 77–105 | MR | Zbl

[3] Akhiezer N. I., Izbrannye trudy po teorii funktsii i matematicheskoi fizike, t. 1, 2, AKTA, Kharkov, 2001

[4] Levin B. Ya., Lectures on entire functions, Transl. Math. Monographs, 150, Amer. Math. Soc., Providence, RI, 1966 | MR

[5] Gorin E. A., “Estimation of a partial involution in a Banach algebra”, Russian J. Math. Phys., 5:1 (1997), 117–118 | MR | Zbl

[6] Rudin U., Funktsionalnyi analiz, Mir, M., 1975 | MR

[7] Norvidas S. T., “Ob ustoichivosti differentsialnykh operatorov v prostranstvakh tselykh funktsii”, Dokl. AN SSSR, 291:3 (1986), 548–551 | MR

[8] Gorin E. A., “Universal symbols on locally compact Abelian groups”, Bull. Polish Acad. Sci., Math., 51:2 (2003), 199–204 | MR | Zbl

[9] Gorin E. A., “Obobschenie odnoi teoremy Fuglede”, Algebra i analiz, 5:4 (1993), 83–97 | MR | Zbl

[10] Vatson G. N., Teoriya besselevykh funktsii, IL, M., 1949

[11] Abramovits M., Stigan I. (red.), Spravochnik po spetsialnym funktsiyam, Nauka, M., 1979 | MR