Question about shift and linear operators

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Alexander Korovyev

Posted: Sat Dec 27, 2003 11:14 am

Guest

Suppose that T1 and T2 are shift operators on R. Is it provable that
if there exists function K such that K o T1 = T2 o K then K is linear.

Thanks!

Rob Johnson

Posted: Sat Dec 27, 2003 12:35 pm

Guest

In article <ukqruv0qjsvoimjlatg53do8l445nk6vql@4ax.com>,
David C. Ullrich <ullrich@math.okstate.edu> wrote:

Quote:

On 27 Dec 2003 08:14:28 -0800, korovyev@rambler.ru (Alexander
Korovyev) wrote:

Suppose that T1 and T2 are shift operators on R. Is it provable that
if there exists function K such that K o T1 = T2 o K then K is linear.

Hmm. This says that there exist constants a and b such that

K(x+a) = K(x) + b

for all x. This certainly does not imply that K is linear, (or even
affine, which is no doubt what you meant). For example,
the restriction of K to the interval [0,a) could be any function
whatever.

If there are a and b so that for all x

K(x+a) = K(x) + b

then K(x) - bx/a is periodic with period a. Thus, K is the sum of a
linear function and a periodic function. This confirms the statement
that K can be anything on [0,a), but K on that interval plus one more
point uniquely determines K.

Rob Johnson <rob@trash.whim.org>
take out the trash before replying

David C. Ullrich

Posted: Sat Dec 27, 2003 3:26 pm

Guest

On 27 Dec 2003 08:14:28 -0800, korovyev@rambler.ru (Alexander
Korovyev) wrote:

Quote:

Suppose that T1 and T2 are shift operators on R. Is it provable that
if there exists function K such that K o T1 = T2 o K then K is linear.

Hmm. This says that there exist constants a and b such that

K(x+a) = K(x) + b

for all x. This certainly does not imply that K is linear, (or even
affine, which is no doubt what you meant). For example,
the restriction of K to the interval [0,a) could be any function
whatever.

Quote:

Thanks!

************************

David C. Ullrich

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