I am trying to investigate following problem (not only to prove it or disprove it in its current form, but also learn a "context" around it).
Let $A, B$ be vector spaces of traceless matrices such that A is orthogonal to B (in Frobenius inner product sense) and $S = \{ a + b\ |\ a \in A, b \in B \}$ is an internal direct sum. Moreover $A, B$ commute, meaning that for $a \in A, b \in B$ we have $ab = ba$.
I am interested in proving (or disproving) following inequality:
$$ \lVert s \rVert_2^2 \leq \lVert a \rVert_2^2 + \lVert b \rVert_2^2 $$ where: $\lVert \cdot \rVert_2$ is matrix 2-norm (its biggest singular value).
We already know that without commuting property the inequality does not hold: Relationship between matrix 2-norm of internal direct sum and orthogonal vector spaces of traceless matrices.
But, while browsing through the math.stackexchange.com I found following post, that asks similar question and gets positive answer. I do not have great grasp on functional analysis, so I might be very well misunderstanding something. But combining following part about direct sum
whenever a Banach space $X$ is the (algebraic) direct sum of two closed subspaces $E$ and $F$, then the natural projections $\pi_E$ and $\pi_F$ are continuous. Therefore, for every $x$ in $E$, the new norm $$ |||x||| = \|\pi_E(x)\| + \|\pi_F(x)\| $$ is equivalent to the original norm on $X$
And this comment about equivalent but not equal norms:
Equivalent norms need not be equal! It is enough that $A\lVert x \rVert \leq \lVert \lvert x \rVert \rvert \leq B \lVert x \rVert$ , for all x . And yes, the norm need not come from an inner product, but if would would like it to be just define it as $|||x|||=(∥π_E(x)∥^2+∥π_F(x)∥^2)^{1/2}$ and everythig else will be just the same.
My questions are as follows:
- Do I need "only" to find projections such that $\pi_E$ and $\pi_F$, such that $\lVert \pi_E(A) \rVert_2 \leq \lVert A \rVert_2, \lVert \pi_F(B) \rVert_2 \leq \lVert B \rVert_2$
- Do I need to find such that $\pi_E$ and $\pi_F$ that $(\lVert \pi_E(A) \rVert_2 + \lVert \pi_F(B) \rVert_2)^2 \leq \lVert A \rVert_2^2 + \lVert B \rVert_2^2$
- Am I missing the mark completely?