Answered: Problem 1: Let A be a Banach algebra.…

Problem 1: Let A be a Banach algebra. Let x, y € A such that xy = e ‡ yx. Show that yx satisfies (yx)² = yx, i.e., yx is idempotent.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Problem 1: Let A be a Banach algebra. Let x, y € A such that xy = e ‡yx. Show that yx
satisfies (yx)² = yx, i.e., yx is idempotent.

Expert Solution

Step 1

Banach Algebra:- A is called a Banach algebra if it satisfy the following axioms

(1) A is a Banach space;
(2) for x,y,z inA and c be a scalar

(xy)z = x(yz),

(x + y)z = xz + yz,

x(y + z) = xy + xz,
c(xy) = (cx)y = x(cy)
Moreover, there is a unit element e:

ex =xe = x for all x in A;
(3) || e ||= 1;
(4) ||xy||≤ ||x|| ||y|| for all x, y in A.

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