Fundamental Symmetries in Krein Spaces

Posted on 2026/07/31 by arkajad

We are back to indefinite metric, Krein spaces, and fundamental symmetries

Fundamental Symmetries in Krein Spaces
Voice of Krein Space

The pdf is here.

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Polar Decomposition and Partial Isometries

Posted on 2026/07/15 by arkajad
Polar bear and Maine Coon cat contemplating a partial isometry 𝑈 : 𝐻 → 𝐻 ′ U:H→H ′ on the snow.
Trying to understand partial isometries: a polar bear, a Maine Coon, and a rainbow arrow from H\mathcal{H} to H′\mathcal{H}’.

In this note I revisit the polar decomposition T=U∣T∣ for bounded (possibly antilinear) operators between Hilbert spaces, emphasizing the role of partial isometries and their initial and final subspaces.

The diagram in the snow shows exactly what our polar bear and cat are contemplating: a partial isometry whose initial subspace sits inside H, and whose final subspace lives inside H‘.

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Monotone convergence for bounded self-adjoint operators

Posted on 2026/07/08 by arkajad

I realized that before the Square Root Lemma post, I should have included a short section. Here is this section.

The following Lemma is often being used in the construction of the square root of a self-adjoint positive operator. We will not use it, but, for completeness, we will state it here in full generality, for self-adjoint and mot necessarily positive operators.

Lemma. [Monotone convergence for bounded self-adjoint operators]
Let \mathcal H be a real or complex Hilbert space and let (A_\iota)_{\iota\in\mathcal I} be a net of
bounded self-adjoint operators on \mathcal H such that

    \[ \iota_1\le \iota_2 \implies A_{\iota_1}\le A_{\iota_2}, \]

with respect to the quadratic-form order (x,Ax)\le (x,Bx) for all x\in\mathcal H.
Assume there exists a bounded self-adjoint operator B with A_\iota\le B for all
\iota\in\mathcal I. Then there is a bounded self-adjoint operator A_0\le B such that

    \[ (x,A_0x) = \sup_{\iota\in\mathcal I} (x,A_\iota x), \qquad x\in\mathcal H, \]

and A_\iota \to A_0 in the strong operator topology. Equivalently: for every x\in \h, for every \epsilon>0, there is \iota_0\in\mathcal{I} such that \Vert A_\iota x-A_0x\Vert\leq \epsilon for all \iota\geq \iota_0.

In particular, if 0\le A_\iota\le B for all \iota, then A_0 is positive.

Proof.
The proof, covering both real and complex cases, can be found in [1, Proposition 13, Ch. V.46]

References.

[1] Bourbaki, n., ”Topological Vector Spaces Chapters 1–5”, Springer 2003.

There will be one more necessary addition: The Polar Decomposition. I am working on putting all this Hilbert space stuff in one big file, for reference.

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