Radial-like Toeplitz operators on Cartan domains of type I

Let $\mathrm{D}^\mathrm{I}_{n \times n}$ be the Cartan domain of type I which consists of the complex $n \times n$ matrices $Z$ that satisfy $Z^*Z < I_n$. For a symbol $a \in L^\infty(\mathrm{D}^\mathrm{I}_{n \times n})$ we consider three radial-like type conditions: 1) left (right) $\mathrm{U}(n)$-invariant symbols, which can be defined by the condition $a(Z) = a\big((Z^*Z)^\frac{1}{2}\big)$ ($a(Z) = a\big((ZZ^*)^\frac{1}{2}\big)$, respectively), and 2) $\mathrm{U}(n) \times \mathrm{U}(n)$-invariant symbols, which are defined by the condition $a(A^{-1}ZB) = a(Z)$ for every $A, B \in \mathrm{U}(n)$. We prove that, for $n \geq 2$, these yield different sets of symbols. If $a$ satisfies 1), either left or right, and $b$ satisfies 2), then we prove that the corresponding Toeplitz operators $T_a$ and $T_b$ commute on every weighted Bergman space. Furthermore, among those satisfying condition 1), either left or right, there exist, for $n \geq 2$, symbols $a$ whose corresponding Toeplitz operators $T_a$ are non-normal. We use these facts to prove the existence, for $n \geq 2$, of commutative Banach non-$C^*$ algebras generated by Toeplitz operators.