Department of Mathematics & Statistics, Kenyon College — Research Talk

Hermitian Projections on Some Operator Spaces Contributed Talk

Something old, something new, something borrowed, something blue
Speaker Priyadarshi Dey
Venue New Faces in Operator Theory & Function Theory, JMM 2025, Seattle, WA
Date January 2025
Based on Botelho, Dey, Ilišević — Linear Algebra Appl., 2020

Abstract

A projection $P$ on a complex Banach space $X$ is called a Hermitian projection if $P + e^{it}(I-P)$ is an isometry for every $t \in \mathbb{R}$. This talk surveys and extends the theory of Hermitian projections on classical operator spaces. Starting from the classical results of Berkson–Sourour and Stachó–Zalar, we present new characterizations of Hermitian projections on $C(K,E)$, $L^p(\Omega,X)$, and — our main result — on $\mathcal{B}(X,Y)$ for ideal pairs $(X,Y)$. The key tools are a theorem of Fong–Sourour on operator identities and the Khalil–Saleh characterization of surjective isometries on $\mathcal{B}(X,Y)$.

Introduction and Motivation

Definition — Hermitian Operator

Let $X$ be a complex Banach space and $T \in \mathcal{B}(X)$. The operator $T$ is Hermitian if $e^{itT}$ is an isometry for every $t \in \mathbb{R}$.

Examples of trivial Hermitian operators: scalar multiples $Tx = \lambda x$ for $\lambda \in \mathbb{R}$.

Spaces with Only Trivial Hermitian Operators

Berkson, Sourour (Studia Math., 1974): Hermitian operators are trivial on:

Example: Not a Hermitian Operator

On $C[0,1]$, consider $H(f)(x) = f(1-x)$. Taking $f(x) = x$: $$e^{2\pi itH}(f)(x) = (\cos 2\pi t)\,x + i\sin 2\pi t\,(1-x).$$ For $t = \frac{1}{8}$: $\|e^{2\pi i\frac{1}{8}H}(f)\| = \frac{\sqrt{2}}{2} \neq \|f\|$. So $H$ is not Hermitian.

Theorem — Jamison (Linear Algebra Appl., 2007)

A projection $P$ on a complex Banach space $X$ is a Hermitian projection if and only if $P + e^{it}(I-P)$ is an isometry for all $t \in \mathbb{R}$.

Note: Every orthogonal projection on a Hilbert space is a Hermitian projection.

Hermitian Projections on Classical Operator Spaces

Theorem — Stachó and Zalar (Linear Algebra Appl., 2004)
  1. A Hermitian projection $P: \mathcal{B(H)} \to \mathcal{B(H)}$ has the form $x \mapsto Qx$ or $x \mapsto xQ$ for some $Q = Q^* = Q^2 \in \mathcal{B(H)}$.
  2. A Hermitian projection $P: \mathcal{S(H)} \to \mathcal{S(H)}$ satisfies $P=0$ or $P=I$.
  3. A Hermitian projection $P: \mathcal{A(H)} \to \mathcal{A(H)}$: there exists a unit vector $\alpha \in \mathcal{H}$ such that either $Px = Qx + xQ^T$ or $(I-P)x = Qx + xQ^T$ where $Q = \alpha \otimes \alpha$.
Proposition — Dey, Botelho, Ilišević

Let $K$ be a compact Hausdorff space, $E$ a Banach space, and $T$ a Hermitian projection on $C(K,E)$. Then there exists a map $t \mapsto A(t)$ from $K$ into $\mathcal{HP}(E)$ such that $$TF(t) = A(t)F(t), \quad \text{for every } F \in C(K,E) \text{ and } t \in K,$$ where $\mathcal{HP}(E)$ denotes the space of all Hermitian projections on $E$.

Proposition — Dey, Botelho, Ilišević

Let $(\Omega,\Sigma,\mu)$ be a finite measure space and $X$ a separable Banach space. A Hermitian operator $T$ on $L^p(\Omega,X)$, $1 \leq p < \infty$, $p \neq 2$, is a projection if and only if there exists a strongly measurable $t \mapsto A(t)$ from $\Omega$ into $\mathcal{HP}(X)$ such that $$(Tf)(t) = A(t)f(t), \quad \text{for every } f \in L^p(\Omega,X) \text{ and } t \in \Omega.$$

Corollary

Let $E$ be a complex symmetric sequence space different from $\ell^2$. Then $T$ on $E$ is a Hermitian projection if and only if there exists $a = (a(k)) \in E$ with all entries in $\{0,1\}$ such that $Tx = ax = (a(k)x(k))$.

Main Result: Hermitian Projections on $\mathcal{B}(X,Y)$

Definition — Ideal Pair (Khalil, Saleh, 2003)

A pair of Banach spaces $(X,Y)$ is an ideal pair if: $X$ and $Y$ are reflexive; $X$ and $Y^*$ are strictly convex; $X^*$ has the approximation property; $\mathcal{K}(X,Y)$ is an $M$-ideal in $\mathcal{B}(X,Y)$.

Example (Cohen, 1973): $(\ell^p, \ell^q)$ is an ideal pair for $1 < p \leq q < \infty$.

Main Theorem — Dey, Botelho, Ilišević (LAA, 2020)

Let $(X,Y)$ be an ideal pair such that $X$ is not isometric to $Y^*$. Then $P: \mathcal{B}(X,Y) \to \mathcal{B}(X,Y)$ is a Hermitian projection if and only if:

  1. Either there exists a Hermitian projection $Q \in \mathcal{B}(X)$ such that $P(A) = AQ$ for every $A \in \mathcal{B}(X,Y)$, or
  2. there exists a Hermitian projection $R \in \mathcal{B}(Y)$ such that $P(A) = RA$ for every $A \in \mathcal{B}(X,Y)$.
Corollary — Hilbert Space Case

Let $\mathcal{H}$, $\mathcal{K}$ be two non-isometric Hilbert spaces. Then $P: \mathcal{B}(\mathcal{H},\mathcal{K}) \to \mathcal{B}(\mathcal{H},\mathcal{K})$ is a Hermitian projection if and only if either $P(A) = AQ$ for some Hermitian projection $Q \in \mathcal{B}(\mathcal{H})$, or $P(A) = RA$ for some Hermitian projection $R \in \mathcal{B}(\mathcal{K})$.

Key Ingredients and Proof Sketch

Theorem — Fong, Sourour (C.J. Math., 1979)

If $\sum_{k=1}^m A_k T B_k = 0$ for all $T \in \mathcal{B}(X,Y)$, then either $\{B_1,\ldots,B_m\}$ is linearly independent and $A_i = 0$ for every $i$, or $\{B_1,\ldots,B_m\}$ is linearly dependent.

Theorem — Khalil, Saleh (Proc. AMS, 2003)

Let $(X,Y)$ be an ideal pair with $X$ and $Y^*$ not isometric. Then $T: \mathcal{B}(X,Y) \to \mathcal{B}(X,Y)$ is a surjective isometry if and only if there are surjective isometries $U \in \mathcal{B}(Y)$ and $V \in \mathcal{B}(X)$ such that $T(f) = UfV$.

Proof Sketch
  1. Given a Hermitian projection $T$ and $t \in (0,1)$, using Khalil–Saleh we obtain $$Tf = \frac{U_t f V_t - e^{it}f}{1 - e^{it}}$$ with $U_t$, $V_t$ surjective isometries on $Y$ and $X$ respectively.
  2. Since $T$ is a projection, for each $t$: $$U_t^2 f V_t^2 - (1+e^{2\pi it}) U_t f V_t + e^{2\pi it} f = 0 \quad \text{for every } f \in \mathcal{B}(X,Y).$$
  3. By Fong–Sourour, $\{V_t^2, V_t, I\}$ is linearly dependent for every $t$, which forces either $V_t = I$ or $U_t = I$, yielding the two cases of the theorem.

References