# On the Structure of Coisometric Extensions

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

- extend A if $\pi A=B\pi $;
- (power) dilate A if ${\pi}^{*}{B}^{n}\pi ={A}^{n}$ for every $n\ge 0$.

**Definition**

**2.**

## 2. Extensions and Dilations

**Remark**

**1.**

**Proposition**

**1.**

**Proof.**

**Remark**

**2.**

**Definition**

**3.**

**Proposition**

**2.**

**Proof.**

## 3. Unitary Extensions

**Theorem**

**1.**

- (a)
- ${\mathcal{H}}_{u}^{V}$ reduces V to a unitary operator ${V}_{u}$;
- (b)
- ${\mathcal{H}}_{s}^{V}$ reduces V to a shift ${V}_{s}$.

**Theorem**

**2.**

**Theorem**

**3.**

- (a)
- There exists a subspace ${\mathcal{K}}_{0}$ of $\mathcal{K}$, which reduces U such that the spectral measures of the unitary operators ${V}_{u}$ and ${U|}_{{\mathcal{K}}_{0}}$ possess identical multiplicity functions;
- (b)
- There exists a subspace ${\mathcal{K}}_{1}$ of $\mathcal{K}$, which contains ${\mathcal{K}}_{0}$, and it is invariant under U such that the shifts ${V}_{s}$ and ${\left(U{|}_{{\tilde{\mathcal{K}}}_{1}}\right)}_{s}$ possess identical multiplicities (here, ${\tilde{\mathcal{K}}}_{1}:={\bigvee}_{n\ge 0}({\mathcal{K}}_{1}\ominus {U}^{n}{\mathcal{K}}_{1})$); equivalently, $dim(ker{V}^{*})=dim({\mathcal{K}}_{1}\ominus U{\mathcal{K}}_{1})$.

**Proof.**

**Theorem**

**4.**

## 4. Restrictions of Backward Shifts

**Theorem**

**5.**

**Proof.**

**Example**

**1.**

**Theorem**

**6.**

## 5. The General Case

**Example**

**2.**

**Example**

**3.**

**Proposition**

**3**

- (a)
- $0\le {A}_{T}\le {I}_{\mathcal{H}}$;
- (b)
- ${T}^{*}{A}_{T}T={A}_{T}$;
- (c)
- $ker{A}_{T}=\{h\in \mathcal{H}\mid {T}^{n}h\to 0\phantom{\rule{4.pt}{0ex}}as\phantom{\rule{4.pt}{0ex}}n\to \infty \}$;
- (d)
- $ker(I-{A}_{T})=\{h\in \mathcal{H}\mid \parallel {T}^{n}h\parallel =\parallel h\parallel \phantom{\rule{4.pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4.pt}{0ex}}n\ge 0\}$.

**Theorem**

**7.**

**Proof.**

**Example**

**4.**

**Remark**

**3.**

- (a)
- There exists a subspace ${\mathcal{E}}_{0}$ of $\mathcal{E}$, which reduces U such that the spectral measures of the unitary operators ${V}_{T}{|}_{{\bigcap}_{n\ge 0}\overline{{A}_{T}^{1/2}{T}^{n}\mathcal{H}}}$ and ${U}^{*}{|}_{{\mathcal{E}}_{0}}$ possess identical multiplicity functions;
- (b)
- There exists a subspace ${\mathcal{E}}_{1}$ of $\mathcal{E}$ that contains ${\mathcal{E}}_{0}$, and it is invariant under ${U}^{*}$ such that the shifts ${\left({V}_{T}\right)}_{s}$ and ${\left({U}^{*}{|}_{{\bigvee}_{n\ge 0}({\mathcal{E}}_{1}\ominus {U}^{*n}{\mathcal{E}}_{1})}\right)}_{s}$ possess identical multiplicities.

**Theorem**

**8.**

**Remark**

**4.**

**Corollary**

**1.**

**Corollary**

**2.**

## 6. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Reed, M.; Simon, B. Methods of Modern Mathematical Physics; Functional Analysis; Academic Press: San Diego, CA, USA, 1980; Volume 1. [Google Scholar]
- Riesz, F.; Sz.-Nagy, B. Functional Analysis; Ungar: New York, NY, USA, 1955. [Google Scholar]
- Rudin, W. Functional Analysis, 8th ed.; International Series in Pure and Applied Mathematics; McGraw-Hill: New York, NY, USA, 1991. [Google Scholar]
- Sz.-Nagy, B. Sur les contractions de l’espace de Hilbert. Acta Sci. Math. Szeged
**1953**, 15, 87–92. [Google Scholar] - Shalit, O.M. Dilation Theory: A Guided Tour. Operator Theory, Functional Analysis and Applications. Oper. Th. Adv. Appl.
**2021**, 282, 551–623. [Google Scholar] - Sz.-Nagy, B.; Foiaș, C.; Bercovici, H.; Kérchy, L. Harmonic Analysis of Operators on Hilbert Space, 2nd ed.; Revised and Enlarged edition; Springer: New York, NY, USA, 2010. [Google Scholar]
- Foiaș, C.; Frazho, A.E. The Commutant Lifting Approach to Interpolation Problems. In Operator Theory: Advances and Applications; Birkhäuser-Verlag: Basel, Switzerland, 1990; Volume 44. [Google Scholar]
- Foiaș, C.; Frazho, A.E.; Gohberg, I.; Kaashoek, M.A. Metric Constrained Interpolation, Commutant Lifting and Systems. In Operator Theory: Advances and Applications; Birkhäuser-Verlag: Basel, Switzerland, 1998; Volume 100. [Google Scholar]
- Rosenblum, M.; Rovnyak, J. Hardy classes and Operator Theory. In Oxford Mathematical Monographs; Oxford University Press: New York, NY, USA, 1985. [Google Scholar]
- Kakihara, Y. Multidimensional second order stochastic processes. In Series on Multivariate Analysis 2; World Scientific Publishing Co., Inc.: River Edge, NJ, USA, 1997. [Google Scholar]
- Muhly, P.S.; Solel, B. Extensions and dilations for C
^{*}-dynamical systems. Operator Theory, Operator Algebras and Applications. Contemp. Math.**2006**, 414, 375–381. [Google Scholar] - Muhly, P.S.; Solel, B. Tensor algebras over C
^{*}-correspondences: Representations, dilations, and C^{*}-envelopes. J. Funct. Anal.**1998**, 158, 389–457. [Google Scholar] [CrossRef] - Wolf, T. Coisometric Extensions. Ph.D. Thesis, University of Iowa, Iowa City, IA, USA, 2013. [Google Scholar]
- Sz.-Nagy, B.; Foiaș, C. Sur les contractions de l’espace de Hilbert. VII. Triangulations canoniques. Fonctions minimum. Acta Sci. Math. Szeged
**1964**, 25, 12–37. [Google Scholar] - Bhattacharjee, M.; Das, B.K. Factors of hypercontractions. J. Oper. Theory
**2021**, 85, 443–462. [Google Scholar] [CrossRef] - Das, B.K.; Sarkar, J.; Sarkar, S. Factorizations of contractions. Adv. Math.
**2017**, 322, 186–200. [Google Scholar] [CrossRef] - Halmos, P.R. Shifts on Hilbert spaces. J. Reine Angew. Math.
**1961**, 208, 102–112. [Google Scholar] [CrossRef] - Halmos, P.R. Introduction to Hilbert Space and the Theory of Spectral Multiplicity, 2nd ed.; Dover Publications: Mineola, NY, USA, 2017. [Google Scholar]
- Foiaș, C. A remark on the universal model for contractions of G. C. Rota. Com. Acad. R. P. Romîne
**1963**, 13, 349–352. (In Romanian) [Google Scholar] - Schäffer, J.J. On unitary dilations of contractions. Proc. Amer. Math. Soc.
**1955**, 6, 322. [Google Scholar] [CrossRef] - Halperin, I. The unitary dilation of a contraction operator. Duke Math. J.
**1961**, 28, 279–289. [Google Scholar] [CrossRef] - Sz.-Nagy, B. On Schäffer’s construction of unitary dilations. Ann. Univ. Budapest Sect. Math.
**1960**, 3, 343–346. [Google Scholar] - Sz.-Nagy, B.; Foiaș, C. Sur les contractions de l’espace de Hilbert. V. Translation bilatérales. Acta Sci. Math. Szeged
**1962**, 23, 106–129. [Google Scholar] - Bercovici, H.; Kérchy, L. Spectral behaviour of C
_{10}-contractions. In Operator Theory Live; Theta Series in Advanced Mathematics; Theta Foundation: Bucharest, Romania, 2010; Volume 12, pp. 17–33. [Google Scholar] - Cassier, G. Generalized Toeplitz operators, restrictions to invariant subspaces and similarity problems. J. Oper. Theory
**2005**, 53, 49–89. [Google Scholar] - Kubrusly, C.S.; Dugal, B.P. Contractions with C
_{·0}direct summands. Adv. Math. Sci. Appl.**2001**, 11, 593–601. [Google Scholar] - Duggal, B.P. On unitary parts of contractions. Indian J. Pure Appl. Math.
**1994**, 25, 1243–1247. [Google Scholar] [CrossRef] - Kubrusly, C.S. Contractions T for which A is a projection. Acta Sci. Math. Szeged
**2014**, 80, 603–624. [Google Scholar] [CrossRef] - Kubrusly, C.S. Hilbert Space Operators: A Problem Solving Approach; Birkhäuser: Boston, MA, USA, 2003. [Google Scholar]
- Kubrusly, C.S.; Vieira, P.C.M. Strong stability for cohyponormal operators. J. Operator Theory
**1994**, 31, 123–127. [Google Scholar] - Kubrusly, C.S.; Vieira, P.C.M.; Pinto, D.O. A decomposition for a class of contractions. Adv. Math. Sci. Appl.
**1996**, 6, 523–530. [Google Scholar] - Wold, H. A Study in the Analysis of Stationary Time Series; Almqvist and Wiksell: Uppsala, Sweden, 1938. [Google Scholar]
- Xuan, A.; Yin, M.; Li, Y.; Chen, X.; Ma, Z. A comprehensive evaluation of statistical, machine learning and deep learning models for time series prediction. In Proceedings of the 7th International Conference on Data Science and Machine Learning Applications (CDMA), Virtual, 1–3 March 2022; pp. 55–60. [Google Scholar]
- Williams, B.M.; Hoel, L.A. Modeling and forecasting vehicular traffic flow as a seasonal ARIMA process: Theoretical basis and empirical results. J. Transp. Eng.
**2003**, 129, 664–672. [Google Scholar] [CrossRef] - Yu, L.; Yu, L.; Wang, J.; Wang, R.; Chen, Z. Cyclostationary modeling for the aerodynamically generated sound of helicopter rotors. Data Mech. Syst. Signal Process.
**2022**, 168, 108680. [Google Scholar] [CrossRef] - Ghalati, M.K.; Nunes, A.; Ferreira, H.; Serranho, P.; Bernardes, R. Texture Analysis and Its Applications in Biomedical Imaging: A Survey. IEEE Rev. Biomed. Eng.
**2022**, 15, 222–246. [Google Scholar] [CrossRef] [PubMed] - Kolmogorov, A.N. Stationary sequences in Hilbert spaces. Bull. Math. Univ. Moskow
**1941**, 2, 1–40. (In Russian) [Google Scholar] - von Neumann, J. Allgemeine eigenwerttheorie hermitischer funktional operatoren. Math. Ann.
**1929**, 102, 49–131. [Google Scholar] [CrossRef] - Brown, A. A version of multiplicity theory. In Topics in Operator Theory; Mathematical Surveys; American Mathematical Society: Providence, RI, USA, 1974; Volume 13, pp. 129–160. [Google Scholar]
- Kubrusly, C.S. An Introduction to Models and Decompositions in Operator Theory; Birkhäuser: Boston, MA, USA, 1997. [Google Scholar]
- Barik, S.; Das, B.K.; Haria, K.; Sarkar, J. Isometric dilations and von Neumann inequality for a class of tuples in the polydisc. Trans. Am. Math. Soc.
**2019**, 372, 1429–1450. [Google Scholar] [CrossRef] - Sarkar, S. Pairs of Commuting Pure Contractions and Isometric dilation. J. Operator Theory
**2023**, accepted. [Google Scholar] - Curto, R.E.; Vasilescu, F.H. Standard Operators Models in the Polydisc I. Indiana Univ. Math. J.
**1993**, 42, 791–810. [Google Scholar] [CrossRef] - Popovici, D. On C
_{0·}multi-contractions having a regular dilation. Studia Math.**2005**, 170, 297–302. [Google Scholar] [CrossRef]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Popovici, D.
On the Structure of Coisometric Extensions. *Axioms* **2023**, *12*, 202.
https://doi.org/10.3390/axioms12020202

**AMA Style**

Popovici D.
On the Structure of Coisometric Extensions. *Axioms*. 2023; 12(2):202.
https://doi.org/10.3390/axioms12020202

**Chicago/Turabian Style**

Popovici, Dan.
2023. "On the Structure of Coisometric Extensions" *Axioms* 12, no. 2: 202.
https://doi.org/10.3390/axioms12020202