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location Rehovot, Israel
age 24
visits member for 2 years, 8 months
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I'm a PhD student in Weizmann Institute, specializing in Probability.


Dec
14
revised Operator arithmetic-harmonic mean inequality with operator-valued weights
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Dec
14
revised Operator arithmetic-harmonic mean inequality with operator-valued weights
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Dec
14
comment Operator arithmetic-harmonic mean inequality with operator-valued weights
@NarutakaOZAWA: Indeed, the reduction of the $\sum_i \Lambda_i \ge 1$ case to the $\sum_i \Lambda_i = 1$ case is an easy exercise. But I don't see immediately how to prove necessity; in particular, I don't see why the inequality for traces implies operator inequality...
Dec
14
comment Operator arithmetic-harmonic mean inequality with operator-valued weights
@NarutakaOZAWA: Thanks a lot, I'll certainly look at that.
Dec
14
comment Operator arithmetic-harmonic mean inequality with operator-valued weights
@YemonChoi: And it's actually fairly elementary, since the $n=2$ case $(\lambda X_1^{-1} + (1-\lambda) X_2^{-1})^{-1} \le \lambda X_1 + (1-\lambda) X_2$ is equivalent to the more obvious $(\lambda + (1-\lambda) X_1^{1/2} X_2^{-1} X_1^{1/2})^{-1} \le \lambda + (1-\lambda) X_1^{-1/2} X_2 X_1^{-1/2}$, and general $n$ follows by induction from $n=2$.
Dec
14
comment Operator arithmetic-harmonic mean inequality with operator-valued weights
@YemonChoi For scalar weights a stronger statement is known: $\left[ \sum_i \lambda_i X_i^{-1} \right]^{-1} \le \sum_i \lambda_i X_i$ in the operator sense.
Dec
13
revised Operator arithmetic-harmonic mean inequality with operator-valued weights
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Dec
13
asked Operator arithmetic-harmonic mean inequality with operator-valued weights
Nov
28
comment Bounded operator on a normed space with empty spectrum
@GeoffRobinson: It's actually a general fact that if we have a sequence of spaces $B_0 \supset B_1 \supset B_2 \supset \dots$ that are complete, respectively, w.r.t. norms $\Vert \cdot \Vert_0 \le \Vert \cdot \Vert_1 \le \Vert \cdot \Vert_2 \le \dots$ and all $B_{k+1}$ are dense in $(B_k, \Vert \cdot \Vert_k)$ then $\bigcap_k B_k$ is dense in $B_0$. In this case take the norms $\Vert T^{-k} (\cdot) \Vert$ on $\mathop{\mathrm{im}} T^k$.
Nov
28
comment Bounded operator on a normed space with empty spectrum
@GeoffRobinson: $\bigcap_n \mathop{\mathrm{im}} T^n$ is dense iff $\mathop{\mathrm{im}} T$ is dense iff $\ker T^\ast = 0$.
Nov
21
comment What is the best reference for Spectral theory?
Rudin's book gives a great background on functional analysis in general, including the spectral theorem and some very basic notions related to Banach algebras, but it I wouldn't say it touches any "true" spectral theory. It's merely a prerequisite.
Nov
13
revised Is an infinite-dimensional “Lebesgue measure” uniquely determined by a set of positive finite measure?
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Nov
13
revised Is an infinite-dimensional “Lebesgue measure” uniquely determined by a set of positive finite measure?
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Nov
13
asked Is an infinite-dimensional “Lebesgue measure” uniquely determined by a set of positive finite measure?
Nov
6
awarded  Enthusiast
Nov
3
comment Regularity of random Fourier series
@IgorRivin: ... For Gaussians there is a well-developed theory that provides estimates based on the metric-entropy-like properties of the canonical metric on the parameter space, culminating in Talagrand's necessary and sufficient condition for boundedness/continuity. For a readable introduction see Adler's "An introduction to continuity, extrema, ...". There is also Talagrand's book "Generic chaining" on that, but I haven't really read it...
Nov
3
comment Regularity of random Fourier series
@IgorRivin: Well, Kolmogorov's criterion is classical, and you can find at least the one-dimensional version in any textbook that touches stochastic processes; in the $n$-dimensional case see, e.g., Theorem 2.23 in Kallenberg's "Foundations of Modern Probability".
Nov
3
comment Regularity of random Fourier series
@IgorRivin: I would rather say speculate that this "middle" is a manifestation of independence of coefficients - positive or negative correlation of, say, polynomial decay should probably give different regularity exponents. But I haven't done the exercise, so I don't know really...
Nov
2
revised Regularity of random Fourier series
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Nov
2
revised Regularity of random Fourier series
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