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Banach J. Math. Anal. 8 (2014), no. 2, 16–29

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Banach J. Math. Anal. 8 (2014), no. 2, 16–29

B

anach

J

ournal of

M

athematical

A

nalysis ISSN: 1735-8787 (electronic)

www.emis.de/journals/BJMA/

MAXIMAL IDEAL SPACE OF SOME BANACH ALGEBRAS AND RELATED PROBLEMS

SUNA SALTAN1∗ AND YASEM˙IN ¨OZEL2 Communicated by M. Abel

Abstract. LetCA(n):=CA(n)(D×D) denote the subspace of functions in the Banach spaceC(n) D×D

which are analytic in the bi-disc D×D. We con- sider the subspaceBzw consisting from the functions f ∈CA(n) which can be represented in the formf(z, w) =g(zw),wheregis a single variable function from the disc algebraCA(D). We prove thatBzw is a Banach algebra under the Duhamel multiplication

(f~g) (zw) = ∂2

∂z∂w

z

Z

0 w

Z

0

f((z−u) (w−v))g(uv)dvdu

and describe its maximal ideal space. We also consider the Hardy type operator f →xy

x

R

0 y

R

0

f(tτ)dτ dtand discuss its some properties.

1 Suleyman Demirel University, Department of Mathematics, 32260, Isparta, Turkey.

E-mail address: [email protected]

2 Suleyman Demirel University, Department of Mathematics, 32260, Isparta, Turkey.

E-mail address: [email protected]

Date: Received: Apr. 5, 2013; Revised: Jun. 18, 2013; Accepted: Aug. 7, 2013.

∗ Corresponding author.

2010Mathematics Subject Classification. Primary 47B47; Secondary 47B38, 46E35.

Key words and phrases. Banach algebra, radical Banach algebra, Duhamel multiplication, quasinilpotent operator, invariant subspace.

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