User jes&#250;s &#225;lvarez - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T20:58:42Z http://mathoverflow.net/feeds/user/19838 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90466/inequalities-between-self-adjoint-operators Inequalities between self-adjoint operators Jesús Álvarez 2012-03-07T14:47:52Z 2012-03-07T19:20:49Z <p>Let $T_s$ ($s\ge0$) be a smooth family of non-negative self-adjoint operators in a separable Hilbert space $H$. Suppose that, for some $C'>C>0$, we have $T_0+Cs^2\le T_s\le T_0+C's^2$ for all $s$ large enough. Let $\psi:[0,\infty)\to(0,1]$ be a smooth bijection with $\psi'&lt;0$. Is there any $K>0$ such that $\psi(T_s)\le\psi(T_0+Ks^2)$ for $s$ large enough?</p> http://mathoverflow.net/questions/89956/maximum-number-of-orthonormal-vectors-contained-in-an-open-cone Maximum number of orthonormal vectors contained in an open cone Jesús Álvarez 2012-03-01T12:21:14Z 2012-03-01T17:00:02Z <p>Let $H$ be a separable Hilbert space, $\Pi:H\to L$ the orthogonal projection to a linear subspace of finite dimension $p$, and $U$ the open cone of vectors $u\in H$ such that $\langle u,\Pi u\rangle>\|u\|^2/\sqrt{2}$. Which is the maximum number of orthonormal vectors contained in $U$?</p> http://mathoverflow.net/questions/83043/on-the-paley-wiener-theorem On the Paley-Wiener theorem Jesús Álvarez 2011-12-09T08:20:25Z 2011-12-11T10:15:34Z <p>Is there any even Schwartz function whose restriction to $[0,\infty)$ is monotone and whose Fourier transform is compactly supported? In other words, is there any entire analytic function satisfying the condition of the Paley-Wiener theorem that is even on the real line and whose restriction to $[0,\infty)$ is monotone?</p> http://mathoverflow.net/questions/89956/maximum-number-of-orthonormal-vectors-contained-in-an-open-cone Comment by Jesús Álvarez Jesús Álvarez 2012-03-01T17:21:37Z 2012-03-01T17:21:37Z That's right. Thank you! http://mathoverflow.net/questions/89956/maximum-number-of-orthonormal-vectors-contained-in-an-open-cone Comment by Jesús Álvarez Jesús Álvarez 2012-03-01T16:06:38Z 2012-03-01T16:06:38Z About the motivation, I arrived to this question studying growth conditions for the eigenvalues of self-adjoint operators with discrete spectrum. Suggestions about this kind of study are also welcome. As an example, it is easy to see that the maximum number is 1 if $p=1$, and it is $\ge 3$ if $p=2$. The dimension of U is not mentioned in the question, but its definition uses a linear subspace of finite dimension.