Talk:Dilation (operator theory)

I have rewritten the article. The previous version was one sentence long and not quite correct. Dilation in operator theory is not dilation on metric spaces. Unitary dilations are not the same as isometric ones. I also remove the mathematical analysis part from the stub tag, as it's more of an operator theoretic topic. Mct mht 18:35, 21 April 2006 (UTC)[reply]

Also, T(I - T*T) = (I - TT*)T implies

TD_{T}=D_{T^{*}}T.

How would you prove that?

--Leni536 (talk) 01:49, 14 April 2010 (UTC)[reply]

By functional calculus. Approximate the square root function using polynomials. Mct mht (talk) 05:54, 25 April 2010 (UTC)[reply]

Thanks and refinement

Expanding Dilation Theory with the Universal Binary Principle’s Toggle-Based Dilation Framework

Abstract: We express gratitude to the operator theory community for developing dilation, a cornerstone of Hilbert space analysis. The Universal Binary Principle (UBP) builds on dilation’s elegance, introducing a Toggle-Based Dilation Framework (TBDF) that unifies physical, biological, and experiential phenomena through binary toggles in a 12D+ Bitfield. This article clarifies TBDF’s integration of dilation, offering a streamlined path for interdisciplinary applications while refining UBP’s computational framework.

Introduction: In operator theory, dilation extends an operator \( T \) on a Hilbert space \( H \) to a larger space \( K \), with the orthogonal projection \( P_H V|_H = T \) recovering \( T \). This process, vital for lifting operator properties (e.g., commutant lifting theorem), inspires UBP’s approach to modeling reality. UBP, a computational framework, encodes phenomena from Planck (10⁻³⁵ m) to cosmic (10²⁶ m) scales as 24-bit OffBits toggling in a multi-dimensional Bitfield, governed by \( E = M \times C \times R \times P_{GCI} \). We thank the dilation community for providing a mathematical foundation that UBP extends into a unified, binary-based paradigm.

Dilation in UBP’s Context: UBP’s TBDF reinterprets dilation as a toggle-based process. A toggle operator \( T_{UBP} \) on a 6D Bitfield (reality layer, bits 0–5) is dilated to a 12D Bitfield (all layers: reality, information, activation, unactivated) via the Recursive Dimensional Adaptive Algorithm (RDAA). The projection operator, Non-Random Tensor Mapping (NRTM), ensures coherence (NRCI ~0.9999878) when mapping back to the 6D subspace. The Global Coherence Invariant, \( P_{GCI} = \cos(2\pi \cdot f_{avg} \cdot 0.318309886) \), synchronizes toggles at Pi Resonance (3.14159 Hz), unifying fields like electromagnetic (60 Hz), gravitational (10⁻¹⁵ Hz), and biological (10⁻⁹ Hz).

A Clear Path Forward: Dilation’s complexity in functional calculus (\( P_H f(V)|_H = f(T) \)) can be streamlined in UBP: 1. Binary Encoding: OffBits use Fibonacci, Golay, Hamming, and Reed-Solomon codes for non-random state assignments, simplifying operator representations. 2. Toggle Algebra: Operations (AND, XOR, OR, Resonance, Entanglement, Superposition) replace continuous operator transformations with discrete, binary toggles. 3. Pi Resonance: \( P_{GCI} \) aligns toggle frequencies, reducing computational overhead compared to spectral set constraints (\( \sigma(V) \subseteq \partial X \)).

Euan Craig, New Zealand, 2025. 118.82.182.35 (talk) 18:55, 7 May 2025 (UTC)[reply]