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Multivariable pseudospectrum in $C^*$-algebras

Alexander Cerjan, Vasile Lauric, Terry A. Loring

Abstract

We look at various forms of spectrum and associated pseudospectrum that can be defined for noncommuting $d$-tuples of Hermitian elements of a $C^*$-algebra. The emphasis is on theoretical calculations of examples, in particular for noncommuting pairs and triple of operators on infinite dimensional Hilbert space. In particular, we look at the universal pair of projections in a $C^*$-algebra, the usual position and momentum operators, and triples of tridiagonal operators.

Multivariable pseudospectrum in $C^*$-algebras

Abstract

We look at various forms of spectrum and associated pseudospectrum that can be defined for noncommuting -tuples of Hermitian elements of a -algebra. The emphasis is on theoretical calculations of examples, in particular for noncommuting pairs and triple of operators on infinite dimensional Hilbert space. In particular, we look at the universal pair of projections in a -algebra, the usual position and momentum operators, and triples of tridiagonal operators.
Paper Structure (12 sections, 23 theorems, 176 equations, 4 figures)

This paper contains 12 sections, 23 theorems, 176 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Symmetries and $*$-homomorphisms
  3. The commuting and essentially commuting cases
  4. Singular values in infinite dimensions
  5. Other forms of multivariable pseudospectrum
  6. The quadratic pseudospectrum and more
  7. Relation between the Clifford and quadratic pseudospectrum
  8. Symmetry in the quadratic pseudospectrum
  9. The $C^{*}$-algebra generated by two projections
  10. A hemisphere as Clifford spectrum
  11. Momentum and Position -- an unbounded example
  12. Calculations for Example \ref{['ex:Three_Toeplitz_type']}

Key Result

Theorem 2.1

Suppose that $\varphi:\mathcal{A}\rightarrow\mathcal{B}$ is a unital $*$-homomorphism between unital $C^{*}$-algebras. If $A_{1},\dots,A_{d}$ are Hermitian elements of $\mathcal{A}$ then If $\varphi$ is one-to-one, then the inclusion in Equation eqn:Clifford_gets_smaller becomes an equality.

Figures (4)

  • Figure 6.1: Plotted together the Clifford and quadratic pseudospectrum for $(U,V_{z})$, as defined in (\ref{['eqn:two-by-two-projections']},) for various values of of $z$. From the top-left to the bottom-right, the values used are $z=1$, $z=0.6$, $z=0$, $z=-0.5$, $z=-1$.
  • Figure 6.2: The Clifford and quadratic pseudospectrum for the universal pair of order-two unitary matrices $(U,V)$ are plotted together here.
  • Figure 7.1: The Clifford spectrum for the three operators in Equation \ref{['eqn:hemisphere_example']}. $b=1.00$ (left); $b=2.00$ (center); $b=2.05$ (right);
  • Figure 7.2: Plots of $e(x,z)=0$ (dotted curves) and $f(x,z)\geq0$ (light blue area, switched to $\leq$ above $z=b$) for $b=1.00$, $b=2.00$, $b=2.05$.

Theorems & Definitions (46)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Lemma 3.1
  • proof
  • Example 3.2
  • Theorem 3.3
  • ...and 36 more