Mark_Hoffman
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 Apr 21 accepted About a relation between isometries Apr 18 revised About a relation between isometries added 2 characters in body 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 $\ell^2$. $t\in[0,2\pi]$