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 →

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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 →

The post Pre-Hilbert space – its Dual and Completion first appeared on Open System - Ark's blog.

Fundamental Symmetries in Minkowski Krein Space

arkajad — Wed, 20 May 2026 09:05:28 +0000

It is always good to have at least one example. I am reading the Preface to A Course in Functional Analysis by John B. Conway (Springer, 1985), where the author notes: Historically, mathematics has gone from the particular to the general—not the … Continue reading →

The post Fundamental Symmetries in Minkowski Krein Space first appeared on Open System - Ark's blog.

Krein spaces – a quantum-theoretical monad method

arkajad — Tue, 12 May 2026 17:41:40 +0000

In the previous post, Krein spaces – first steps, we introduced the basic definitions and simple examples of Krein spaces, emphasizing how a fundamental symmetry turns an indefinite inner product into a Hilbert space structure. In this post we move from … Continue reading →

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Krein spaces – first steps

arkajad — Sat, 09 May 2026 17:35:10 +0000

It was the summer of 1964 (15 June–15 July). The small village of Katsiveli, in the Yalta region of Crimea, hosted the Second Summer School of Mathematics organized by the Ukrainian Academy of Sciences. Second Summer School of Mathematics, Katsiveli, … Continue reading →

The post Krein spaces – first steps first appeared on Open System - Ark's blog.

Counting infinities

arkajad — Sun, 03 May 2026 15:57:25 +0000

As I have mentioned in my previous post, recently I stepped upon the path leading to infinity. But what is “infinity?” The question demands deep thinking. This post starts the new series. Prolog We usually meet the ideas of finite … Continue reading →

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Infinite geometry

arkajad — Fri, 24 Apr 2026 12:23:34 +0000

A week ago I asked Perplexity AI: “Is there a monograph, or a comprehensive review, discussing infinite-dimensional symmetric spaces and Kahler manifolds?” Here is the answer: Yes. There are a few substantial sources on infinite-dimensional symmetric spaces, and many standard … Continue reading →

The post Infinite geometry first appeared on Open System - Ark's blog.

Recovery

arkajad — Sun, 22 Mar 2026 10:20:49 +0000

This post is an attempt to explain my recent long silence. Over the past few months I have been in a kind of “construction period,” slowly rebuilding my working environment and trying to create a space where I can think … Continue reading →

The post Recovery first appeared on Open System - Ark's blog.