The Hardy Class $mathbbH_2^1times n$
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The Hardy space $mathbbH_2(U)$ on the upper half-plane $U$ is defined to be the space of holomorphic functions $f$ on $U$ with bounded norm,
$$ |f|_H_2 = sup_y>0 left ( int|f(x+ iy)|^2, mathrmdx right)^frac12$$
The Hardy space $mathbbH^ntimes 1_2(U)$ is the Hardy space of $ntimes 1$ matrix valued functions with entries in the space $mathbbH_2(U)$.
$(mathbbH^ntimes 1_2(U))^bot$ is its orthogonal complement :=$f: f^#in mathbbH^1times n_2(U)$ where $f^#(z):=f^*(barz)$
The Schur class $S^ntimes n_in$ is the class of inner holomorphic $ntimes n$ matrix functions on $U$ and satisfy the condition $I_n-s^*(z)s(z)geq 0$.
The problem is the following:
Given $sin S^ntimes n_in$ with $det(s)neq 0$.
Is it true that $f^#(s^-1)^#in mathbbH^1times n_2(U)$ for every $f$ such that $f^#in mathbbH^1times n_2(U)$?
*Is the problem related to the Smirnov maximum principle?
Thanks in advance!
matrices functional-analysis operator-theory
asked Aug 26 at 19:33
John
293
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up vote
2
down vote
favorite
The Hardy space $mathbbH_2(U)$ on the upper half-plane $U$ is defined to be the space of holomorphic functions $f$ on $U$ with bounded norm,
$$ |f|_H_2 = sup_y>0 left ( int|f(x+ iy)|^2, mathrmdx right)^frac12$$
The Hardy space $mathbbH^ntimes 1_2(U)$ is the Hardy space of $ntimes 1$ matrix valued functions with entries in the space $mathbbH_2(U)$.
$(mathbbH^ntimes 1_2(U))^bot$ is its orthogonal complement :=$f: f^#in mathbbH^1times n_2(U)$ where $f^#(z):=f^*(barz)$
The Schur class $S^ntimes n_in$ is the class of inner holomorphic $ntimes n$ matrix functions on $U$ and satisfy the condition $I_n-s^*(z)s(z)geq 0$.
The problem is the following:
Given $sin S^ntimes n_in$ with $det(s)neq 0$.
Is it true that $f^#(s^-1)^#in mathbbH^1times n_2(U)$ for every $f$ such that $f^#in mathbbH^1times n_2(U)$?
*Is the problem related to the Smirnov maximum principle?
Thanks in advance!
matrices functional-analysis operator-theory
asked Aug 26 at 19:33
John
293
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
The Hardy space $mathbbH_2(U)$ on the upper half-plane $U$ is defined to be the space of holomorphic functions $f$ on $U$ with bounded norm,
$$ |f|_H_2 = sup_y>0 left ( int|f(x+ iy)|^2, mathrmdx right)^frac12$$
The Hardy space $mathbbH^ntimes 1_2(U)$ is the Hardy space of $ntimes 1$ matrix valued functions with entries in the space $mathbbH_2(U)$.
$(mathbbH^ntimes 1_2(U))^bot$ is its orthogonal complement :=$f: f^#in mathbbH^1times n_2(U)$ where $f^#(z):=f^*(barz)$
The Schur class $S^ntimes n_in$ is the class of inner holomorphic $ntimes n$ matrix functions on $U$ and satisfy the condition $I_n-s^*(z)s(z)geq 0$.
The problem is the following:
Given $sin S^ntimes n_in$ with $det(s)neq 0$.
Is it true that $f^#(s^-1)^#in mathbbH^1times n_2(U)$ for every $f$ such that $f^#in mathbbH^1times n_2(U)$?
*Is the problem related to the Smirnov maximum principle?
Thanks in advance!
matrices functional-analysis operator-theory
asked Aug 26 at 19:33
John
293
The Hardy space $mathbbH_2(U)$ on the upper half-plane $U$ is defined to be the space of holomorphic functions $f$ on $U$ with bounded norm,
$$ |f|_H_2 = sup_y>0 left ( int|f(x+ iy)|^2, mathrmdx right)^frac12$$
The Hardy space $mathbbH^ntimes 1_2(U)$ is the Hardy space of $ntimes 1$ matrix valued functions with entries in the space $mathbbH_2(U)$.
$(mathbbH^ntimes 1_2(U))^bot$ is its orthogonal complement :=$f: f^#in mathbbH^1times n_2(U)$ where $f^#(z):=f^*(barz)$
The Schur class $S^ntimes n_in$ is the class of inner holomorphic $ntimes n$ matrix functions on $U$ and satisfy the condition $I_n-s^*(z)s(z)geq 0$.
The problem is the following:
Given $sin S^ntimes n_in$ with $det(s)neq 0$.
Is it true that $f^#(s^-1)^#in mathbbH^1times n_2(U)$ for every $f$ such that $f^#in mathbbH^1times n_2(U)$?
*Is the problem related to the Smirnov maximum principle?
Thanks in advance!
matrices functional-analysis operator-theory
asked Aug 26 at 19:33
John
293
asked Aug 26 at 19:33
John
293
asked Aug 26 at 19:33
John
293
asked Aug 26 at 19:33
John
293
293
add a comment |Â
add a comment |Â
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