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.



Is there an abstract theory of multi-spectral radii?
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](https://mathoverflow.net/questions/323445/spectral-radius-for-multiple-linear-operators).


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](https://mathoverflow.net/a/424866/22277) or [this link](https://circcashcore.com/2022/07/24/lower-dimensional-approximations-to-the-l_2-spectral-radius/) 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](https://circcashcore.com/2022/07/24/lower-dimensional-approximations-to-the-l_2-spectral-radius/), and [here is another page](https://circcashcore.com/2022/08/08/measuring-block-cipher-round-function-security-using-spectral-radii/) 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](https://circcashcore.com/2023/02/28/empirical-observations-of-l_2d-spectral-radii-and-other-functions/).


**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.


