# Is there an abstract theory of multi-spectral radii?

There seems to be many valid ways of generalizing the notion of the spectral radius $$\rho(A)$$ of a complex matrix $$A$$ to spectral radii of multiple operators. I am wondering if there is an abstract theory of what it means to be a multi-spectral radius $$\rho(A_1,\dots,A_r)$$ of complex matrices $$A_1,\dots,A_r$$.

Example 0: Suppose that $$A_1,\dots,A_r$$ are complex matrices and $$1\leq p<\infty$$. Then define $$\rho_p(A_1,\dots,A_r)=\lim_{n\rightarrow\infty}\big(\sum_{i_1,\dots,i_n\in\{1,\dots,r\}}\|A_{i_1}\dots A_{i_r}\|^p\big)^{1/(pn)}.$$

We can call this notion of the spectral radius the $$L_p$$-spectral radius.

Theorem: $$\rho_2(A_1,\dots,A_r)^2=\rho(A_1\otimes\overline{A_1}+\dots+A_r\otimes\overline{A_r})$$. Here, $$\overline{A}=(A^*)^T=(A^T)^*$$. Alternatively, define the completely positive linear mapping $$\Phi(A_1,\dots,A_r):M_n(\mathbb{C})\rightarrow M_n(\mathbb{C})$$ by setting $$\Phi(A_1,\dots,A_r)(X)=A_1XA_1^*+\dots+A_rXA_r^*$$. Then $$\rho_2(A_1,\dots,A_r)^2=\rho(\Phi(A_1,\dots,A_r))$$.

It seems like $$\rho_2(A_1,\dots,A_r)$$ is the best way of generalizing the notion of a spectral radius to multiple operators if I had to choose one notion of a spectral radius.

Example 1: Let $$\mathfrak{k}=(k_n)_{n=0}^{\infty}$$ be a sequence of numbers in the set $$\{1,\dots,r\}$$. Then define $$\rho_\mathfrak{k}(A_1,\dots,A_r)=\limsup_{n\rightarrow\infty}\|A_{k_0}\dots A_{k_n}\|^{1/n}.$$

Example 2: Define $$\rho_{2,1}(A_1,\dots,A_r)$$ to be the largest value of $$(|z_1^2|+\dots+|z_r^2|)^{-1/2}$$ where $$I-(z_1A_1+\dots+z_rA_r)$$ is not invertible. Then $$\rho_{2,1}$$ is another generalized notion of a spectral radius. More generally, if $$\|\cdot\|$$ is a complex norm on $$\mathbb{C}^n$$, then define $$\rho_{2,\|\cdot\|}(A_1,\dots,A_r)$$ to be the largest value of $$\|(z_1,\dots,z_r)\|^{-1}$$ where $$I-(z_1A_1+\dots+z_rA_r)$$ is not invertible.

If $$\rho$$ is a multi-spectral radius function, then it seems like $$\rho$$ should satisfy properties such as log-plurisubharmonicity, continuity, homogeneity of degree 1, invariance under joint-similarity, and a few mundane properties. For some particularly nice multi-spectral radius functions $$\rho$$, the value $$\rho(A_1,\dots,A_r)$$ only depends on the completely positive superoperator $$\Phi(A_1,\dots,A_r)$$. And if a multi-spectral radius function $$\rho$$ only depends on the superoperator $$\Phi(A_1,\dots,A_r)$$, then I would imagine that this multi-spectral radius easily generalizes to bounded operators between Hilbert spaces.

What would be a good axiomatization for multi-spectral radius functions? Is there an axiomatization for what is meant by a multi-spectrum of multiple operators? Clearly, one may consider the spectrum of $$\Phi(A_1,\dots,A_r)$$ as a multi-spectrum of $$A_1,\dots,A_r$$. However, if $$z_1,\dots,z_r$$ are complex numbers with $$|z_1|^2+\dots+|z_r|^2=1$$ and where $$\rho(z_1A_1+\dots+z_rA_r)$$ is maximized, then the spectrum $$\sigma(z_1A_1+\dots+z_rA_r)$$ may be considered as another notion of a multi-spectrum.

I am interested in these generalized spectral radii since I have originally used a generalized notion of the spectral radius for cryptocurrency research and development, but such generalized spectral radii seem applicable for other machine learning applications.

• Not really an answer to the question, but you might be interested in looking up "joint spectral radius" and "generalized spectral radius". Commented Apr 19, 2023 at 2:07
• Dixon & Müller investigate a generalisation of nilpotence which may be of interest: matwbn.icm.edu.pl/ksiazki/sm/sm102/sm10237.pdf Commented Apr 19, 2023 at 9:32

I claim that there is a somewhat abstract notion of a multi-spectral radius and that there is probably an abstract theory behind this abstract notion. I will try to justify this abstract multi-spectral radius by showing that it captures the specific examples of multi-spectral radii that I have mentioned in the question and that the simplest examples of these multi-spectral radii are reasonable mathematical objects. With that being said, there are notions of a multi-spectral radius that I have not shown to fit within this framework, so more research on this topic is needed.

Suppose that $$A$$ is a complex Banach algebra. We say that a function $$\rho:A^r\rightarrow[0,\infty)$$ is a multi-spectral radius if there is an isometric embedding $$\iota:A\rightarrow B$$ along with a bounded subset $$\mathcal{C}\subseteq B^r$$ where

1. $$x_j\iota(a)=\iota(a)x_j$$ whenever $$a\in A,(x_1,\dots,x_r)\in\mathcal{C}$$,

2. if $$(x_1,\dots,x_r)\in\mathcal{C}$$, then $$(\lambda_1x_1,\dots,\lambda_rx_r)\in\mathcal{C}$$ whenever $$|\lambda_j|=1$$ for $$1\leq j\leq r$$, and

3. $$\rho(a_1,\dots,a_r)=\rho_{\iota,\mathcal{C}}(a_1,\dots,a_r)=\sup_{(x_1,\dots,x_r)\in\mathcal{C}}\rho(x_1\iota(a_1)+\dots+x_r\iota(a_r))$$ whenever $$a_1,\dots,a_r\in A$$.

One may also want to require that if $$(x_1,\dots,x_r)\in\mathcal{C}$$, then $$(\lambda_1x_1,\dots,\lambda_rx_r)\in\mathcal{C}$$ whenever $$|\lambda_j|=1$$ for $$1\leq j\leq r$$, but this condition is not necessary. One can show that if $$a,b\in B$$, then the mapping from $$\mathbb{C}$$ to $$[-\infty,\infty)$$ defined by $$\lambda\mapsto \ln(\rho(a+\lambda b))$$ is subharmonic, so by the maximum principle,

$$\max\{\rho(\lambda_1x_1\iota(a_1)+\dots+\lambda_rx_r\iota(a_r)):|\lambda_1|=\dots=|\lambda_r|=1\}=\max\{\rho(\lambda_1x_1\iota(a_1)+\dots+\lambda_rx_r\iota(a_r)):|\lambda_1|\leq 1,\dots,|\lambda_r|\leq 1\}.$$

The $$L_1$$-spectral radius can be characterized in terms of our framework.

Theorem: $$\rho_1(a_1,\dots,a_r)$$ is the maximum value of $$\rho(x_1\iota(a_1)+\dots+x_r\iota(a_r))$$ where $$\iota:A\rightarrow B$$ is an isometric embedding of Banach algebras, and $$\|x_j\|\leq 1$$ for $$1\leq j\leq r$$.

The proof of the above result is not too hard, and I have given a proof of the above result in this answer.

We say that a multi-spectral radius $$\rho$$ is unitary invariant if $$\rho(a_1,\dots,a_r)=\rho(b_1,\dots,b_r)$$ whenever there is an $$n\times n$$-unitary matrix $$(u_{i,j})_{i,j}$$ where $$b_j=\sum_{i=1}^ru_{i,j}a_i$$ for $$1\leq j\leq r$$. The following lemma is a standard result from quantum information theory.

Lemma: Suppose that $$A_1,\dots,A_r,B_1,\dots,B_r\in M_n(\mathbb{C})$$. Then $$\Phi(A_1,\dots,A_r)=\Phi(B_1,\dots,B_r)$$ if and only if there is an $$r\times r$$-unitary matrix $$(u_{i,j})_{i,j}$$ where $$B_j=\sum_{i=1}^ru_{i,j}A_i$$ for $$1\leq j\leq r$$.

Therefore, a multi-spectral radius $$\rho:M_n(\mathbb{C})^r\rightarrow[0,\infty)$$ is unitary invariant if and only if $$\rho(A_1,\dots,A_r)=\rho(B_1,\dots,B_r)$$ whenever $$\Phi(A_1,\dots,A_r)=\Phi(B_1,\dots,B_r)$$. The continuous unitary invariant multi-spectral radii are completely determined by the mapping $$\Phi(A_1,\dots,A_r)\mapsto\rho(A_1,\dots,A_r)$$ where $$\Phi(A_1,\dots,A_r)$$ is completely positive and trace preserving (a completely positive trace preserving map is known as a quantum channel).

The following easy lemmas show that how we can always upgrade a multi-spectral radius to a unitary invariant multi-spectral radius.

Lemma: Let $$A$$ be an algebra over a field $$K$$. Suppose that $$(a_1,\dots,a_r),(b_1,\dots,b_r),(x_1,\dots,x_r),(y_1,\dots,y_r)\in A^r$$. Let $$(u_{i,j})_{i,j},(v_{i,j})_{i,j}\in M_r(K)$$ be inverse matrices. Suppose that $$a_k=\sum_{i=1}^ru_{i,k}b_i$$ and $$x_k=\sum_{j=1}^rv_{k,j}y_j$$ for $$1\leq k\leq r$$. Then $$\sum_{k=1}^ra_kx_k=\sum_{i=1}^rb_iy_j.$$

Lemma: Suppose that $$K$$ is a field and $$A$$ is an algebra over $$K$$. Let $$(u_{i,k})_{i,k}\in M_r(K)$$. Suppose furthermore that $$a_1,\dots,a_r,b_1,\dots,b_r,x_1,\dots,x_r,y_1,\dots,y_r\in A$$ and $$a_k=\sum_{i=1}^ru_{i,k}b_i$$ for $$1\leq k\leq r$$ and $$x_i=\sum_{k=1}^ru_{i,k}y_k$$ for $$1\leq k\leq r$$. Then $$\sum_{k=1}^ra_ky_k=\sum_{i=1}^rb_ix_i$$.

Proposition: Let $$A,B$$ be Banach algebras. Let $$\iota:A\rightarrow B$$ be an isometric embedding. Suppose that $$\mathcal{C}\subseteq B^r$$ is a bounded subset with $$x_j\iota(a)=\iota(a)x_j$$ for $$1\leq j\leq r$$. Let $$\mathcal{D}$$ be the collection of all tuples $$(y_1,\dots,y_r)$$ where there is some $$r\times r$$-unitary matrix $$(u_{i,j})_{i,j}$$ and $$(x_1,\dots,x_r)\in\mathcal{C}$$ where $$y_j=\sum_{i=1}^ru_{i,j}x_i$$ for $$1\leq i\leq r$$. Then $$\rho_{\iota,\mathcal{D}}(x_1,\dots,x_r)=\sup\{\rho_{\iota,\mathcal{C}}(\sum_ju_{1,j}x_j,\dots,\sum_ju_{r,j}x_j)\mid (u_{i,j})_{i,j}\in U(r)\}.$$

By using the following version of Holder's inequality that can be proven using the classical Holder's inequality, we can show that the $$L_2$$-spectral radius is a multi-spectral radius.

Theorem: $$\rho(A_1\otimes B_1+\dots+A_r\otimes B_r)\leq \rho_p(A_1,\dots,A_r)\cdot\rho_q(B_1,\dots,B_r)$$ whenever $$p,q\in(1,\infty)$$ and $$\frac{1}{p}+\frac{1}{q}=1$$.

As a consequence, if $$d\geq n$$ and $$A_1,\dots,A_r\in M_n(\mathbb{C})$$, then $$\rho_2(A_1,\dots,A_r)=\max_{(X_1,\dots,X_r)\in M_d(\mathbb{C})}\frac{\rho(A_1\otimes X_1+\dots+A_r\otimes X_r)}{\rho_2(X_1,\dots,X_r)}.$$

We have another construction that allows us to show that the $$L_p$$-spectral radius is a multi-spectral radius for $$1\leq p<\infty$$. Suppose now that $$1\leq p<\infty$$. Now, let $$A$$ be a Banach algebra. Let $$x_1,\dots,x_r$$ be non-commutating variables. Let $$B$$ be the collection of all sums of the form $$\sum_{k=0}^n\sum_{i_1,\dots,i_k\in\{1,\dots,r\}}a_{i_1,\dots,i_k}x_{i_1}\dots x_{i_k}$$. We observe that for $$p>1$$ the Banach space $$\ell^p$$ indexed with the natural numbers cannot be endowed with a convolution operation since $$(1/n)_{n=1}^{\infty}*(1/n)_{n=1}^{\infty}=(+\infty)_{n=1}^\infty$$. We can give $$B$$ a norm that combines the $$\ell^p$$ and the $$\ell^1$$ norms that makes the completion of $$B$$ into a Banach algebra.

Then give $$B$$ the norm $$\|\sum_{k=0}^n\sum_{i_1,\dots,i_k\in\{1,\dots,r\}}a_{i_1,\dots,i_k}x_{i_1}\dots x_{i_k}\|=\sum_{k=0}^n\|(a_{i_1,\dots,i_k})_{i_1,\dots,i_k}\|_p.$$ Give $$B$$ the multiplication defined by bilinearity along with the condition that $$(a\cdot x_{i_1}\dots x_{i_m})\cdot (b\cdot x_{j_1}\dots x_{j_n})= ab\cdot x_{i_1}\dots x_{i_m}x_{j_1}\dots x_{j_n}.$$ In other words, each element in $$A$$ commutes with each variable $$x_j$$, but we do not impose any other version of commutativity.

$$B$$ is submultiplicative: Let $$u=\sum_{j=0}^\infty\sum_{i_1,\dots,i_j\in\{1,\dots,r\}}a_{i_1,\dots,i_j}x_{i_1}\dots x_{i_j}$$ and let $$v=\sum_{j=0}^\infty\sum_{i_1,\dots,i_j\in\{1,\dots,r\}}a_{i_1,\dots,i_j}x_{i_1}\dots x_{i_j}$$ where only finitely many terms of these 'non-commutative polynomials' are non-zero.

Then

$$\|u\cdot v\|$$ $$=\|(\sum_{k=0}^\infty\sum_{i_1,\dots,i_k\in\{1,\dots,r\}}a_{i_1,\dots,i_k}x_{i_1}\dots x_{i_k})\cdot (\sum_{k=0}^\infty\sum_{i_1,\dots,i_k\in\{1,\dots,r\}}b_{i_1,\dots,i_k}x_{i_1}\dots x_{i_k})\|$$

$$=\|\sum_{k=0}^{\infty}\sum_{j=0}^k\sum_{i_1,\dots,i_j\in\{1,\dots,r\}}\sum_{i_{j+1},\dots,i_k}a_{i_1,\dots,i_j}b_{i_{j+1},\dots,i_k}x_{i_1}\dots x_{i_k}\|$$

$$\leq\sum_{k=0}^{\infty}\sum_{j=0}^k\|\sum_{i_1,\dots,i_k\{1,\dots,r\}}a_{i_1,\dots,i_j}b_{i_{j+1},\dots,i_k}x_{i_1}\dots x_{i_k}\|$$

$$=\sum_{k=0}^{\infty}\sum_{j=0}^k\|(a_{i_1,\dots,i_j}\cdot b_{i_{j+1},\dots, i_k})_{i_1,\dots,i_k\in\{1,\dots,r\}}\|_p$$ $$\leq\sum_{k=0}^{\infty}\sum_{j=0}^k\|(a_{i_1,\dots,i_j})_{i_1,\dots,i_j\in\{1,\dots,r\}}\|_p\cdot \|(b_{i_{j+1},\dots,i_k})_{i_{j+1},\dots,i_k\in\{1,\dots,r\}}\|_p$$ $$=\sum_{j=0}^\infty\|(a_{i_1,\dots,i_j})_{i_1,\dots,i_j\in\{1,\dots,r\}}\|_p\cdot\sum_{k=0}^\infty\|(b_{i_1,\dots,i_k})_{i_1,\dots,i_k\in\{1,\dots,r\}}\|_p=\|u\|\cdot\|v\|.$$

Therefore, the completion $$\overline{B}$$ of $$B$$ is a Banach algebra, and the original Banach algebra $$A$$ embeds into $$\overline{B}$$. In this case, we simply have $$\rho_p(a_1,\dots,a_r)=\rho(a_1x_1+\dots+a_rx_r)$$.

One should be able to generalize the above construction to most sensible notions of a multi-spectral radius.

Other examples:

In order for our notion of a multi-spectral radius to be sensible, one would expect that the functions $$\rho_{\iota,\mathcal{C}}$$ would be coherent and interesting for the simplest possible cases of $$\iota,\mathcal{C}$$. For example, if $$\iota:A\rightarrow A$$ is the identity function and $$1\leq p\leq\infty$$, and $$\mathcal{C}$$ is the unit ball in $$\mathbb{C}^r$$ with respect to the $$p$$-norm, then one should expect for $$\rho_{\iota,\mathcal{C}}$$ to be about as reasonable of a function as the $$L_p$$-spectral radii, and experimental computations indicate that this is indeed the case.

Define a mapping $$F_{\iota,\mathcal{C},a_1,\dots,a_r}:\mathcal{C}\rightarrow[0,\infty)$$ by $$F_{\iota,\mathcal{C},a_1,\dots,a_r}(x_1,\dots,x_r)= \rho(x_1\iota(a_1)+\dots+x_r\iota(a_r))$$. Experimental computations suggest that the local maxima $$(x_1,\dots,x_r)$$ for the function $$F_{\iota,\mathcal{C},a_1,\dots,a_r}$$ tend to resemble a sort of conjugate of $$a_1,\dots,a_r$$.

Let $$\iota_n:M_n(\mathbb{C})\rightarrow M_n(\mathbb{C})$$ be the identity mapping, and let $$\mathcal{L}_{r;p}=\{(\lambda_1,\dots,\lambda_r)\in \mathbb{C}^r:\|(\lambda_1,\dots,\lambda_r)\|_p=1\}$$.

In some of my experiments with $$A_1,\dots,A_r\in M_n(\mathbb{R})$$ and in all of my experiments with $$A_1,\dots,A_r\in M_n(\mathbb{C})$$ that are Hermitian or real symmetric, when I computed $$\lambda_1,\dots,\lambda_r$$ locally maximizes $$F_{\iota_n,S_1^r,A_1,\dots,A_r}$$, then one can find a $$\lambda\in S_1$$ and $$e_1,\dots,e_r\in\{-1,1\}$$ where $$\lambda_j=\lambda\cdot e_j$$ for $$1\leq j\leq r$$. A similar phenomenon holds when I locally maximized $$F_{\iota_n,\mathcal{L}_{r;p},A_1,\dots,A_r}$$ for $$1\leq p\leq\infty$$ even though this phenomenon seems to break down as $$p$$ gets close to $$1$$ and it holds better for Hermitian matrices than it does for non-symmetric real matrices.

My computer experiments indicate that if we locally maximize $$F_{\iota,\mathcal{C},a_1,\dots,a_r}$$, then as $$\mathcal{C}$$ better approximates $$A$$, the local maxima $$(x_1,\dots,x_r)$$ will become more and more similar to a conjugate version of $$(a_1,\dots,a_r)$$. On the other hand, if $$\mathcal{C}$$ is too complicated and has too much room to work with, then the local maxima $$(x_1,\dots,x_r)$$ will again poorly represent the elements in $$A$$. Therefore, in order to best represent the conjugates of the elements in $$A$$, it is best if $$\mathcal{C}$$ is a little bit simpler than $$A$$.

Let $$\iota_{r;n,d}:M_n(\mathbb{C})\rightarrow M_{n\times d}(\mathbb{C})$$ be the algebra homomorphism defined by $$\iota_{r;n,d}(A)=A\otimes I_d$$. Let $$\mathcal{C}_{r;n,d}$$ be the collection of all tuples $$(I_n\otimes X_1,\dots,I_n\otimes X_r)$$ where $$\rho_2(X_1,\dots,X_r)=1$$.

Theorem: Suppose that $$A_1,\dots,A_r,B_1,\dots,B_r$$ are $$n\times n$$-complex matrices where $$A_1,\dots,A_r$$ do not have a common invariant subspace. Suppose furthermore that $$\rho_2(A_1,\dots,A_r)>0,\rho_2(B_1,\dots,B_r)>0$$. Then $$\rho(A_1\otimes B_1+\dots+A_r\otimes B_r)=\rho_2(A_1,\dots,A_r)\rho_2(B_1,\dots,B_r)$$ if and only if there is some $$\lambda$$ and invertible $$C$$ where $$B_j=\overline{\lambda\cdot C\cdot A_j\cdot C^{-1}}$$ for $$1\leq j\leq r$$.

See this answer or this link for proofs that I gave of the above result.

From the above result, we see that if $$I_n\otimes\overline{X_1},\dots,I_n\otimes\overline{X_r}\in M_{n\times n}(\mathbb{C})$$ globally maximizes $$F_{\iota_{r;n,n},\mathcal{C}_{r;n,n},A_1,\dots,A_r}$$ and $$A_1,\dots,A_r$$ have no common invariant subspace, then there are $$C,\lambda$$ where $$X_j=\lambda CA_jC^{-1}$$ for $$1\leq j\leq r.$$

If $$A_1,\dots,A_r\in M_n(\mathbb{C})$$ does not have a common invariant subspace and $$I_n\otimes\overline{X_1},\dots,I_n\otimes\overline{X_r}\in M_{n\times d}(\mathbb{C})$$ locally maximizes $$F_{\iota_{r;n,d},\mathcal{C}_{r;n,d},A_1,\dots,A_r}$$, then the matrices $$X_1,\dots,X_r$$ will (up-to-similarity and a constant factor) resemble $$A_1,\dots,A_r$$. For example, if $$A_1,\dots,A_r$$ are all real, complex symmetric, real symmetric, Hermitian, real positive semidefinite, complex positive semidefinite, quaternionic, rank $$\leq k$$, etc, and $$I_n\otimes\overline{X_1},\dots,I_n\otimes\overline{X_r}$$ locally maximizes $$F_{\iota_{r;n,d},\mathcal{C}_{r;n,d},A_1,\dots,A_r}$$, then one will often be able to find a constant $$\lambda$$ and invertible matrix $$C$$ where $$Y_j=\lambda CX_rC^{-1}$$ satisfy those properties respectively. Furthermore, one will often be able to find matrices $$R,S$$ where $$Y_j=RA_jS$$ for $$1\leq j\leq r$$. In this case, $$RS=I_d$$ and $$P=SR$$ will be a (non-orthogonal) projection matrix. Define linear operators $$F,G:M_n(\mathbb{C})\rightarrow M_n(\mathbb{C})$$ by setting $$F(X)=\sum_{k=1}^rA_kX(PA_kP)^*$$ and $$G(X)=\sum_{k=1}^rA_k^*XPA_kP$$ (here $$F=G^*$$). Define $$U_0=I_n,V_0=I_n$$ and set $$U_{n+1}=F(U_n)/\|F(U_n)\|,V_{n+1}=G(V_n)/\|G(V_n)\|$$ for $$n\geq 0$$. Then $$U_n,V_n$$ experimentally converge to positive semidefinite matrices $$U,V$$, the dominant eigenvectors of $$F$$ and $$G$$. It seems like the strategy that the optimization algorithm chose for locally maximizing the spectral radius was to make the dominant eigenvalues of $$F,G$$ positive semidefinite matrices of rank $$d$$, but the best way to retain the positive semidefiniteness of the dominant eigenvectors of $$F,G$$ is to make the operators $$PA_kP$$ closely related to the operators $$A_k$$. It seems like the reason this strategy works is that in order for a spectral radius of a matrix $$A$$ to be large, the matrix $$A$$ should be designed to maximize a particular eigenvalue, and by making the operators $$PA_kP$$ related to $$A_k$$, we can maximize the spectral radius of $$F,G$$. Since the local maximum values of $$F_{\iota_{r;n,d},\mathcal{C}_{r;n,d},A_1,\dots,A_r}$$ are closely related to the tuples $$(A_1,\dots,A_r)$$ themselves, I would regard the multi-spectral radius $$\rho_{\iota_{r;n,d},\mathcal{C}_{r;n,d}}$$ as a legitimate generalization of the notion of the spectral radius to multiple operators which I call the $$L_{2,d}$$-spectral radius $$\rho_{2,d}$$.

Other multi-spectral radii $$\rho_{\iota,\mathcal{C}}$$ are probably reasonably well-behaved, but more computer experiments are needed to verify whether other multi-spectral radii $$\rho_{\iota,\mathcal{C}}$$ behave nearly as well as $$\rho_{\iota_{r;n,d},\mathcal{C}_{r;n,d}}$$.

One can find more details on $$\rho_{\iota_{r;n,d},\mathcal{C}_{r;n,d}}$$ at my site here, and here is another page where I apply $$\rho_{\iota_{r;n,d},\mathcal{C}_{r;n,d}}$$ to evaluate cryptographic algorithms. I also gave some experimental observations of $$\rho_{\iota_{r;n,d},\mathcal{C}_{r;n,d}}$$ right here.

Multi-spectrum:

There seems to be a somewhat reasonable definition of a multi-spectrum of a collection of operators.

Suppose that $$\rho_{\iota,\mathcal{C}}$$ is a multi-spectral radius. If $$(x_1,\dots,x_r)\in\mathcal{C}$$ and $$\rho(x_1\iota(a_1)+\dots+x_r\iota(a_r))=\rho_{\iota,\mathcal{C}}(a_1,\dots,a_r)$$, then we say that the spectrum of $$x_1\iota(a_1)+\dots+x_r\iota(a_r)$$ is a multi-spectrum of $$a_1,\dots,a_r$$ with respect to the embedding $$\iota$$ and set $$\mathcal{C}$$.

Suppose that $$(x_1,\dots,x_r)\in\mathcal{C}$$ and for every neighborhood $$U$$ of $$(x_1,\dots,x_r)$$ with respect to the topology induced by the norm on $$A$$, then whenever $$(y_1,\dots,y_r)\in U\cap\mathcal{C}$$, we have $$\rho(x_1\iota(a_1)+\dots+x_r\iota(a_r))\geq\rho(y_1\iota(a_1)+\dots+y_r\iota(a_r))$$; then we say that the spectrum of $$x_1\iota(a_1)+\dots+x_r\iota(a_r)$$ is a local multi-spectrum of $$(a_1,\dots,a_r)$$ with respect to the embedding $$\iota$$ and the set $$\mathcal{C}$$.

This notion of a multi-spectrum depends on the choice of $$\iota,\mathcal{C}$$ and not only on the multi-spectral radius $$\rho_{\iota,\mathcal{C}}$$. For example, if $$A_1,\dots,A_r$$ are complex matrices with no common invariant subspace, then the multi-spectrum of $$A_1,\dots,A_r$$ with respect to $$\iota_{r;n,n}$$ and $$\mathcal{C}_{r;n,n}$$ is simply the spectrum of $$A_1\otimes\overline{A_1}+\dots+A_r\otimes\overline{A_r}$$. On the other hand, suppose that $$x_1,\dots,x_r$$ are the non-commuting variables in $$\overline{B}$$ where $$\rho_p(A_1,\dots,A_r)=\rho(x_1\iota(A_1)+\dots+x_r\iota(A_r))$$ for all matrices $$A_1,\dots,A_r$$ and suppose that $$\mathcal{C}=\{(\lambda_1 x_1,\dots,\lambda_r x_r)\mid \lambda_1,\dots,\lambda_r\in S_1\}$$. Then a multi-spectrum of $$(A_1,\dots,A_r)$$ with respect to $$\iota$$ and $$\mathcal{C}$$ is the spectrum of $$x_1A_1+\dots+x_rA_r$$. If $$\lambda$$ is complex number with $$|\lambda|=1$$, then there is an automorphism $$\phi$$ of the Banach algebra $$B$$ with $$\phi(x_1A_1+\dots+x_rA_r)=\lambda(x_1A_1+\dots+x_rA_r)$$. Therefore, since $$x_1A_1+\dots+x_rA_r$$ has the same spectrum as $$\lambda(x_1A_1+\dots+x_rA_r)$$, there is some compact set $$C\subseteq[0,\infty)$$ with $$\sigma(x_1A_1+\dots+x_rA_r)=\{\lambda t:|\lambda|=1,t\in C\}$$.

Unresolved properties

If the set $$\mathcal{C}$$ is compact, then the function $$\rho_{\iota,\mathcal{C}}$$ is automatically upper-semicontinuous. If $$\rho_{\iota,\mathcal{C}}$$ is not upper-semicontinuous, then we can just take the upper-semicontinuous regularization of $$\rho_{\iota,\mathcal{C}}$$, but the possible lack of upper-semicontinuity is a potential problem with the abstract theory that I am proposing. I do not know if we should require $$\mathcal{C}$$ to always be compact in order to make $$\rho_{\iota,\mathcal{C}}$$ always upper-semicontinuous.

So far, I have mainly experimental results about multi-spectral radii, but I would like for there to be more theorems about multi-spectral radii.