How and why to solve the operator equation $AX-XB=Y$ TeX Export

@article{citeulike:816662, abstract = { Let \$A\$ be an operator on Hilbert space. Question: When (i.e., under what conditions on \$A\$) can we be sure that the operator equation \$AX=Y\$ will have a unique solution \$X\$ for every possible \$Y\$? Answer: Exactly when \$A\$ is invertible. Translation: When the spectrum of \$A\$, \$\sigma(A)\$, does not contain \$0\$. The answer is the same if the equation is \$XB=Y\$ instead (with the fixed operator on the right instead of the left). Since operators seldom commute, there is a third kind of linear equation involving operators: \$AX-XB=Y\$. What conditions on \$A\$ and \$B\$ guarantee the existence of a unique solution \$X\$ for each \$Y\$? (A note of warning: you need to read the third paragraph of this paper to figure out what operators are supposed to be fixed, and which ones varying.) The authors trace the study of this problem to Sylvester in 1884; the best-known extension to operators is by Rosenblum, so they refer to the following fact as the Sylvester-Rosenblum theorem: If \$\sigma(A) \cap \sigma(B) = \emptyset\$, then the equation \$AX-XB=Y\$ has a unique solution \$X\$ for every \$Y\$. The discussion of this result is barely longer in the paper than it is in this review. The bulk of the paper is a discussion of other parts of mathematics in which the Sylvester-Rosenblum theorem arises. A number of applications to other parts of operator theory are taken up: similarity theory (e.g., when are \$\left(\smallmatrix A \& C \\0 \& B \endsmallmatrix\right)\$ and \$\left(\smallmatrix A \& 0 \\0 \& B \endsmallmatrix\right)\$ similar?), hyperinvariant subspaces (e.g., when is an invariant subspace hyperinvariant?), spectral operators, Lyapunov's equation, and Embry's commutativity theorem. <P> All of these application depend just on the existence of a solution in the Sylvester-Rosenblum theorem. Of course, the title asks "how" to solve the equation, not just "is there a solution", and the authors take up the issue of actually finding a solution; approaches to this issue fall into two categories, those involving infinite series, and those expressing the solution as an integral. If \$A\$ and \$B\$ are normal, Hermitian, or unitary, a number of simplifications can be made. Once constructive solutions are available, the authors consider questions relating to the size (norm) of the solution, with applications to Fourier transforms. Finally, some questions relating to perturbation theory are brought up. For example, suppose that Hermitian operators \$A\$ and \$B\$ are "close" (in norm). Do the spectral measures of \$A\$ and \$B\$ have to be "close"? In general, they do not, but certain results are available. One result is the Davis-Kahan "\$\sin\theta\$" theorem, familiar to numerical analysts. Bhatia and Rosenthal show how the Davis-Kahan theorem can be proved using the Sylvester-Rosenblum theorem. Another example: If \$A\$ and \$B\$ are invertible and close, how close are the unitary factors in the polar decompositions? <P> The results in this paper are not intended to be new; the purpose is to show how the simple and well-known Sylvester-Rosenblum theorem can flex its muscles in so many arenas, many of them unfamiliar to operator theorists (at least to this one). The paper is nicely written; it tells us succinctly how to prove the basic theorems, and enough about the applications to make us want to look up more details.}, author = {Bhatia, Rajendra and Rosenthal, Peter}, citeulike-article-id = {816662}, citeulike-linkout-0 = {http://dx.doi.org/10.1112/S0024609396001828}, citeulike-linkout-1 = {http://blms.oxfordjournals.org/cgi/content/abstract/29/1/1}, citeulike-linkout-2 = {http://www.ams.org/mathscinet-getitem?mr=1416400}, doi = {10.1112/S0024609396001828}, journal = {Bull. London Math. Soc.}, keywords = {formal\_theory, linear\_algebra}, mrnumber = {MR1416400}, number = {1}, pages = {1--21}, posted-at = {2007-07-16 14:58:25}, priority = {2}, title = {How and why to solve the operator equation \$AX-XB=Y\$}, url = {http://dx.doi.org/10.1112/S0024609396001828}, volume = {29}, year = {1997} }

TY - JOUR ID - citeulike:816662 L3 - citeulike-article-id:816662 N2 - Let $A$ be an operator on Hilbert space. Question: When (i.e., under what conditions on $A$) can we be sure that the operator equation $AX=Y$ will have a unique solution $X$ for every possible $Y$? Answer: Exactly when $A$ is invertible. Translation: When the spectrum of $A$, $\sigma(A)$, does not contain $0$. The answer is the same if the equation is $XB=Y$ instead (with the fixed operator on the right instead of the left). Since operators seldom commute, there is a third kind of linear equation involving operators: $AX-XB=Y$. What conditions on $A$ and $B$ guarantee the existence of a unique solution $X$ for each $Y$? (A note of warning: you need to read the third paragraph of this paper to figure out what operators are supposed to be fixed, and which ones varying.) The authors trace the study of this problem to Sylvester in 1884; the best-known extension to operators is by Rosenblum, so they refer to the following fact as the Sylvester-Rosenblum theorem: If $\sigma(A) \cap \sigma(B) = \emptyset$, then the equation $AX-XB=Y$ has a unique solution $X$ for every $Y$. The discussion of this result is barely longer in the paper than it is in this review. The bulk of the paper is a discussion of other parts of mathematics in which the Sylvester-Rosenblum theorem arises. A number of applications to other parts of operator theory are taken up: similarity theory (e.g., when are $\left(\smallmatrix A & C \\0 & B \endsmallmatrix\right)$ and $\left(\smallmatrix A & 0 \\0 & B \endsmallmatrix\right)$ similar?), hyperinvariant subspaces (e.g., when is an invariant subspace hyperinvariant?), spectral operators, Lyapunov's equation, and Embry's commutativity theorem. <P> All of these application depend just on the existence of a solution in the Sylvester-Rosenblum theorem. Of course, the title asks "how" to solve the equation, not just "is there a solution", and the authors take up the issue of actually finding a solution; approaches to this issue fall into two categories, those involving infinite series, and those expressing the solution as an integral. If $A$ and $B$ are normal, Hermitian, or unitary, a number of simplifications can be made. Once constructive solutions are available, the authors consider questions relating to the size (norm) of the solution, with applications to Fourier transforms. Finally, some questions relating to perturbation theory are brought up. For example, suppose that Hermitian operators $A$ and $B$ are "close" (in norm). Do the spectral measures of $A$ and $B$ have to be "close"? In general, they do not, but certain results are available. One result is the Davis-Kahan "$\sin\theta$" theorem, familiar to numerical analysts. Bhatia and Rosenthal show how the Davis-Kahan theorem can be proved using the Sylvester-Rosenblum theorem. Another example: If $A$ and $B$ are invertible and close, how close are the unitary factors in the polar decompositions? <P> The results in this paper are not intended to be new; the purpose is to show how the simple and well-known Sylvester-Rosenblum theorem can flex its muscles in so many arenas, many of them unfamiliar to operator theorists (at least to this one). The paper is nicely written; it tells us succinctly how to prove the basic theorems, and enough about the applications to make us want to look up more details. IS - 1 JF - Bull. London Math. Soc. EP - 21 TI - How and why to solve the operator equation $AX-XB=Y$ VL - 29 SP - 1 KW - formal_theory KW - linear_algebra AU - Bhatia, Rajendra AU - Rosenthal, Peter PY - 1997/// UR - http://dx.doi.org/10.1112/S0024609396001828 ER -