Assume $A, B$ are self-ajoint compact operators. Is it true that $\|A+iB\|\le \|2A+iB\|$? Do we have a stronger inequality $\prod_{k=1}^ns_k(A+iB)\le \prod_{k=1}^ns_k(2A+iB)$ or even stronger one $s_n(A+iB)\le s_n(2A+iB)$, $n=1, 2, \ldots$, where $s_n$ are s-numbers?
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$\begingroup$ Even the first inequality doesn't seem to hold - take $2A + iB = 0$ for non-zero $A$. $\endgroup$– robibokDec 27, 2012 at 15:40
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$\begingroup$ they are self-ajoint $\endgroup$– BetrandDec 27, 2012 at 15:48
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1$\begingroup$ could you add some motivation behind the first inequality? $\endgroup$– SuvritDec 27, 2012 at 19:54
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$\begingroup$ A motivation is from the scalar case, as in this article math.pku.edu.cn/teachers/yaoy/Fall2011/Fan_Hoffman1955.pdf $\endgroup$– BetrandDec 27, 2012 at 22:17
2 Answers
I just tried a few random matrices... $$ \begin{align} &A=\begin{bmatrix} -0.1 & -0.4\cr -0.4&0\end{bmatrix}, \qquad B=\begin{bmatrix} 1.5 & -0.5 + i\cr -0.5 - i& 3.5\end{bmatrix}, \cr &\|A+iB\| = \frac{7}{2}+\frac{\sqrt{61}}{10}\approx 4.28 \cr &\|2A+iB\| = \frac{7}{2}+\frac{\sqrt{29}}{10}\approx 4.04 \end{align} $$
$s_n(A+iB) \not\le s_n(2A+iB)$, for example by setting \begin{equation*} A = \begin{bmatrix} -2 & -6\\\\ -6 & -2 \end{bmatrix}\qquad B = \begin{bmatrix} 10 & 4\\\\ 4 & 16 \end{bmatrix}. \end{equation*} In this case, $s_2(A+iB) = 7.7192$ and $s_2(2A+iB) = 4.9433$.
$\prod_{k=1}^n s_k(A+iB) \not\le \prod_{k=1}^n s_k(2A+iB)$ for the same example as above. The lhs is 156, while the rhs is 129.2440.
The operator norm version also does not hold (as shown in the nice counterexample by Gerald Edgar)
EDIT
If, however, $A$ and $B$ are assumed to be positive definite, then these inequalities probably hold. As a hint why they might hold (I have not had the time to check any of the other cases), consider $C=A+iB$ and $D=2A+iB$ with $A,B \ge 0$. Then,
\begin{equation*} \begin{split} \prod_{j=1}^n s_j(C) = |\det C| &= \det(A)\prod_{j=1}^n[1+ s_j(A^{-1/2}BA^{-1/2})^2]^{1/2}\\\\ \prod_{j=1}^n s_j(D) = |\det D| &= \det(A)\prod_{j=1}^n2\left[1+ \frac{s_j(A^{-1/2}BA^{-1/2})^2}{4}\right]^{1/2}\\\\ &= \det(A)\prod_{j=1}^n[4+ s_j(A^{-1/2}BA^{-1/2})^2]^{1/2} \ge |\det(C)|. \end{split} \end{equation*}
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$\begingroup$ Thanks, but how do you define determinant? If $A, B$ are positive definite matrices, we have $|\det(2A+iB)|\ge |\det(A+iB)|$; see Lemma 5 of Kh. D. Ikramov, Determinantal inequalities for accretive-dissipative matrices, J. Math. Sci. (N. Y.), 121(2004) 2458-2464. $\endgroup$– BetrandDec 28, 2012 at 19:29
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$\begingroup$ I have shown above the determinant that you claim in your comment. The absolute value of the determinant of a complex matrix $X$ is the product of its singular values. Using that, my edit shows that $|\det(2A+iB)| \ge |\det(A+iB)|$, which of course, is probably the most trivial result of its type :-) $\endgroup$– SuvritDec 28, 2012 at 20:19