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Jesús Álvarez
Jesús Álvarez
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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 $\psi'<0$. Is there someany $K>0$ such that $\psi(T_s)\le\psi(T_0+Ks^2)$ for $s$ large enough?

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 $\psi'<0$. Is there some $K>0$ such that $\psi(T_s)\le\psi(T_0+Ks^2)$ for $s$ large enough?

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 $\psi'<0$. Is there any $K>0$ such that $\psi(T_s)\le\psi(T_0+Ks^2)$ for $s$ large enough?

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Jesús Álvarez
Jesús Álvarez
  • 329
  • 2
  • 9

Inequalities between self-adjoint operators

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 $\psi'<0$. Is there some $K>0$ such that $\psi(T_s)\le\psi(T_0+Ks^2)$ for $s$ large enough?