Open System – Ark's blog

Polar Decomposition and Partial Isometries

arkajad — Wed, 15 Jul 2026 17:56:22 +0000

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 … Continue reading →

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Monotone convergence for bounded self-adjoint operators

arkajad — Wed, 08 Jul 2026 18:09:02 +0000

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 … Continue reading →

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Functoriality of the Borel Functional Calculus and Pushforward of Spectral Measures

arkajad — Sat, 04 Jul 2026 17:08:56 +0000

This post is just the final part of Spectral integrals and a first taste of functional calculus. The pdf file is here.

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Spectral integrals and a first taste of functional calculus

arkajad — Fri, 03 Jul 2026 10:11:45 +0000

Today it just a pdf document. Available here. Afternotes. 04-07-26 Added property (iii) in the defining properties of a spectral measure Definition 2). I have forgotten about this important property in the original version.

The post Spectral integrals and a first taste of functional calculus first appeared on Open System - Ark's blog.

Spectral family and resolution of the identity

arkajad — Thu, 25 Jun 2026 10:43:51 +0000

This post is (hopefully) the last one in a short series on real and complex Hilbert spaces. After this post we will return to the study of real and complex spaces with indefinite metric, Krein spaces and their geometry. In … Continue reading →

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Square root lemma

arkajad — Wed, 17 Jun 2026 08:35:51 +0000

This post is a continuation of Positive operators II, We begin by recalling the main fact proved there, then explain how a classical binomial series leads to a functional calculus for the square root of a positive operator. Corollary 1 … Continue reading →

The post Square root lemma first appeared on Open System - Ark's blog.

Positive operators II

arkajad — Thu, 11 Jun 2026 17:12:01 +0000

This post is a continuation of Positive operators I. The aim here is to explain how the norm of a bounded self–adjoint operator can be expressed in terms of the associated quadratic form. We will first recall the Riesz representation … Continue reading →

The post Positive operators II first appeared on Open System - Ark's blog.

Positive operators I

arkajad — Thu, 04 Jun 2026 17:41:16 +0000

I realized that I have to diverge even more on my way to Krein spaces. We will need square roots of positive operators on real and complex Hilbert spaces, an so there is a need to discuss this subject in … Continue reading →

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Complexification of a Real Krein Space. Part I

arkajad — Mon, 01 Jun 2026 17:08:16 +0000

Introduction There is a rather unusual journal called the Journal of Humanistic Mathematics. It aims to provide an open forum for academic and informal discussions of the “human face of mathematics,” focusing on aesthetic, cultural, historical, literary, pedagogical, philosophical, psychological, … Continue reading →

The post Complexification of a Real Krein Space. Part I first appeared on Open System - Ark's blog.

Pre-Hilbert space – its Dual and Completion

arkajad — Fri, 29 May 2026 16:18:18 +0000

This post is something of an interlude. I was supposed to be writing about complexification: how to turn real vector spaces into complex ones, and what that means for quantum mechanics. In the complex setting, the standard formulation of quantum mechanics is … Continue reading →

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