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1d
accepted A relation between two properties of sequences of operators
1d
comment A relation between two properties of sequences of operators
Yeah, you are right, I put one more square than necessary :) Thanks a lot!!
1d
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?
1d
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 $B(\ell^2)$ :/
1d
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.
1d
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: $||\sum_{l=1}^{N}T_l\;e^{ilt}||=\sup_{||x||_{\ell^2}=1}||(\sum_{l=1}^{N}T_l\;e^{‌​ilt})(x)||$
1d
revised A relation between two properties of sequences of operators
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1d
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 $T_l$
1d
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.
1d
comment A relation between two properties of sequences of operators
@user1952009 $x$ denotes a sequence in $\ell^2$. $t\in[0,2\pi]$
1d
revised A relation between two properties of sequences of operators
added 9 characters in body
1d
comment A relation between two properties of sequences of operators
Edited the op with an explanation.
1d
revised A relation between two properties of sequences of operators
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2d
comment A relation between two properties of sequences of operators
I'll write that implication when i get home. About the other thing, the operator is not acting on the exponential; $\sum_lT_le^{ilt}$ is the "function" whose Fourier coefficients are those operators $T_l$.
2d
asked A relation between two properties of sequences of operators
Dec
2
comment Hölder type inequality
Thanks a lot for the answer, Milind!
Dec
2
accepted Hölder type inequality
Dec
1
asked Hölder type inequality
Oct
21
awarded  Popular Question
Jul
7
comment About a property of the upper triangular projection of a matrix
In the proof, one of the parts is seeing that A does not belong to $B(\ell^2)$. In order to see this, the author states that $\infty=\lVert{I-P_T(A)}\rVert\leq \lVert{A}\rVert$. I already saw that $\infty=\lVert{I-P_T(A)}\rVert$. The inequality that remains is my doubt, I'm not sure if it is a general property or only something specific for this matrix...