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Apr
21
accepted About a relation between isometries
Apr
18
revised About a relation between isometries
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Apr
18
comment About a relation between isometries
This question might be silly, but what is the meaning for you of $\leq$ when operators are involved?
Apr
13
comment About a relation between isometries
The book I was checking out, where the proposed question comes from is "Analysis and Probability wavelets, signals, fractals" by Jorgensen. The question can be found in page 174, it's the 8,2.
Apr
13
revised About a relation between isometries
added 18 characters in body
Apr
13
comment About a relation between isometries
That's what I was thinking, and why I am confused. But the question comes from the relations known as cuntz relations.en.wikipedia.org/wiki/Cuntz_algebra In a book I saw a proposed question where it says that relation 1 + the fact that $(T_i)_i $are isometries gives automatically relation 2.
Apr
12
asked About a relation between isometries
Apr
7
revised Hadamard product involving operators
edited tags
Apr
7
asked Hadamard product involving operators
Feb
29
awarded  Popular Question
Feb
9
accepted A relation between two properties of sequences of operators
Feb
9
comment A relation between two properties of sequences of operators
Yeah, you are right, I put one more square than necessary :) Thanks a lot!!
Feb
9
comment A relation between two properties of sequences of operators
If $\alpha_l=e^{ilt}$, its modulus is 1, so you would get $\bigg\| \sum_{\ell=1}^N \alpha_\ell T_\ell \bigg\|_{B(\ell^2)} =\sup_{||x||=1}\sum_{l=1}^{N}|x_l|^2<\infty$, right?
Feb
9
comment A relation between two properties of sequences of operators
Thanks for the answer PhoemueX! However, for the second part, this statement you do $$ \bigg\| \sum_{\ell=1}^N \alpha_\ell T_\ell \bigg\| = \|(\alpha_\ell)_{\ell =1 , \dots,N}\|_{\ell^2} $$ (that is, calculating the norm of the function as the norm 2 of the sequence of coefficients) can only be done if the function involved takes values in a Hilbert space, which is not the case. In other words, we can't apply Plancherel if the function takes values in $B(\ell^2)$ :/
Feb
9
comment A relation between two properties of sequences of operators
This is not an exerciSe. Thanks a lot for your comments, if you want me to clear more things, I will try. If someone else has more ideas, I will be very glad to hear them.
Feb
9
comment A relation between two properties of sequences of operators
Everything here is standard notation. I don't know if the confusion here is that $||\cdot||$ denotes both the norm in the space $\ell^2$ and the operator norm...To clear things: $||\sum_{l=1}^{N}T_l\;e^{ilt}||=\sup_{||x||_{\ell^2}=1}||(\sum_{l=1}^{N}T_l\;e^{‌​ilt})(x)||$
Feb
9
revised A relation between two properties of sequences of operators
added 27 characters in body
Feb
9
comment A relation between two properties of sequences of operators
I explained that to PhoemueX above. With $||T_l(x)||$ we are calculating the norm of al alement of $\ell^2$. With $||\sum_{l=1}^N T_l\;e^{ilt}||$, we are calculating the norm of the function whose fourier coefficients are the operators $T_l$
Feb
9
comment A relation between two properties of sequences of operators
In statement A, there is a supremum for all $x$ in the unit sphere of $\ell^2$. In statement b, we are integrating over t, so I don't see the misunderstanding.
Feb
9
comment A relation between two properties of sequences of operators
@user1952009 $x$ denotes a sequence in $\ell^2$. $t\in[0,2\pi]$