## African Diaspora Journal of Mathematics

- Afr. Diaspora J. Math. (N.S.)
- Volume 11, Number 2 (2011), 98-131.

### The Polar Decomposition in Banach Spaces

T. L. Gill, V. Steadman, and W. W. Zachary

#### Abstract

In this paper we survey research progress related to the existence of an adjoint for linear operators on Banach spaces. We introduce a new pair separable Banach spaces which are required for the general theory. We then discuss a number ways one can explicitly construct an adjoint and then prove that one always exists for bounded linear operators. However, this is not true for the class of closed densely defined linear operators. In this case, we can only show that one exists for operators of Baire class one. The existence of an adjoint allows us to construct the polar decomposition. As applications, we extend the Poincaré inequality and the Stone-von Neumann version of the spectral theorem to all operators of Baire class one on a separable Banach space. Our results even show that the spectral theorem is natural for Hilbert spaces (in a certain well-defined sense). As a final application, we provide the natural Banach space version of the Schatten class of compact operators.

#### Article information

**Source**

Afr. Diaspora J. Math. (N.S.), Volume 11, Number 2 (2011), 98-131.

**Dates**

First available in Project Euclid: 6 December 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.adjm/1323180289

**Mathematical Reviews number (MathSciNet)**

MR2862568

**Zentralblatt MATH identifier**

1267.47001

**Subjects**

Primary: 46B03: Isomorphic theory (including renorming) of Banach spaces

Secondary: 47D03: Groups and semigroups of linear operators {For nonlinear operators, see 47H20; see also 20M20} 47H06: Accretive operators, dissipative operators, etc. 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47)

**Keywords**

Poincaré inequality spectral theorem semigroups vector measures vector-valued functions Schatten-class

#### Citation

Gill, T. L.; Steadman, V.; Zachary, W. W. The Polar Decomposition in Banach Spaces. Afr. Diaspora J. Math. (N.S.) 11 (2011), no. 2, 98--131. https://projecteuclid.org/euclid.adjm/1323180289