Multiplicative Spectrum Preservers

• Multiplicative spectrum preservers are maps that preserve the spectral properties of operator products, forcing a rigid classification into automorphic, anti-automorphic, or Jordan forms.
• They employ techniques such as rank-one operator characterization, symmetry conditions, and metric isometry arguments to establish precise structural results.
• These classifications have practical implications in operator algebras, Hilbert-space theory, and stochastic matrix analysis by linking spectral invariants with underlying algebraic symmetries.

Searching arXiv for the cited work on multiplicative spectrum preservers and closely related operator-algebraic classifications. Multiplicative spectrum preservers are maps determined by identities that compare the spectrum, or the peripheral spectrum, of multiplicative expressions before and after transformation. In the operator-algebraic and matrix-theoretic settings represented here, the defining constraints take forms such as

$$\sigma\bigl(\varphi(x)\varphi(y)\bigr)=\sigma(xy),\qquad \sigma_\pi(\Phi(A_1)*\cdots *\Phi(A_k))=\sigma_\pi(A_1*\cdots *A_k),$$

or analogous equalities for skew products, power-type products, and products of stochastic matrices. A recurrent conclusion is rigidity: despite the weakness of the spectral data being preserved, the map is forced into a narrow class of automorphic, anti-automorphic, Jordan, unitary, anti-unitary, transpose, or permutation-implemented forms (Zhang et al., 2013, Taghavi et al., 2013, Mori et al., 2024, Tsai et al., 26 Sep 2025).

1. Definitions and ambient settings

The literature treats several related notions under the heading of multiplicative spectrum preservation. In standard operator algebras on complex Banach spaces, one fixes a generalized product

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$

where each $i_j\in\{1,\ldots,k\}$, every index appears at least once, and at least one index appears exactly once; the integer $m$ is the width of the product. The associated spectral invariant in Zhang–Hou is the peripheral spectrum

$$\sigma_\pi(A):=\{\lambda\in \sigma(A):|\lambda|=r(A)\},$$

where $r(A)$ is the spectral radius. In unital $C^*$-algebras, Mori–Oi consider surjective maps between self-adjoint parts $A_{sa}\to B_{sa}$ satisfying $\sigma(\varphi(x)\varphi(y))=\sigma(xy)$. In Hilbert-space operator algebras, Taghavi–Hosseinzadeh study surjections preserving the spectrum of products such as $A^rBA^s$, $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$0, and $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$1. In stochastic-matrix theory, multiplicative spectrum preservers are lists of maps $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$2 satisfying

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$3

for all inputs in $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$4 (Zhang et al., 2013, Taghavi et al., 2013, Mori et al., 2024, Tsai et al., 26 Sep 2025).

Setting	Preserved quantity	Classified form
Standard operator algebras on Banach spaces	$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$5 of generalized products	$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$6 or $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$7
Hilbert spaces, skew generalized products	$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$8 of skew products	$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$9 or $i_j\in\{1,\ldots,k\}$0
Self-adjoint parts of unital $i_j\in\{1,\ldots,k\}$1-algebras	$i_j\in\{1,\ldots,k\}$2	$i_j\in\{1,\ldots,k\}$3
$i_j\in\{1,\ldots,k\}$4 and $i_j\in\{1,\ldots,k\}$5	$i_j\in\{1,\ldots,k\}$6, $i_j\in\{1,\ldots,k\}$7, $i_j\in\{1,\ldots,k\}$8	$i_j\in\{1,\ldots,k\}$9
$m$0 and their spans	spectrum or trace of $m$1-fold products	permutation or transpose-permutation forms

These settings use different ambient categories, but they all formulate multiplicativity through a product and impose spectral equality on every admissible product. This suggests that the central problem is not merely linear preserver theory, but the interaction between multiplicative structure and spectral invariants.

2. Peripheral-spectrum preservation on standard operator algebras

For standard operator algebras $m$2 and $m$3 on complex Banach spaces, Zhang–Hou prove a classification theorem for maps $m$4 whose range contains all operators of rank at most two and which satisfy

$m$5

for every $m$6-tuple. The theorem states that this condition is equivalent to one of two possibilities. Either there exists an invertible $m$7 and a root of unity $m$8 with $m$9 such that

$$\sigma_\pi(A):=\{\lambda\in \sigma(A):|\lambda|=r(A)\},$$0

or $$\sigma_\pi(A):=\{\lambda\in \sigma(A):|\lambda|=r(A)\},$$1 are reflexive, the index-sequence satisfies the quasi-semi-Jordan symmetry, and

$$\sigma_\pi(A):=\{\lambda\in \sigma(A):|\lambda|=r(A)\},$$2

If the product is not quasi-semi-Jordan, only the first case occurs (Zhang et al., 2013).

The quasi-semi-Jordan condition is imposed on the fixed index-sequence $$\sigma_\pi(A):=\{\lambda\in \sigma(A):|\lambda|=r(A)\},$$3. Writing the unique index as $$\sigma_\pi(A):=\{\lambda\in \sigma(A):|\lambda|=r(A)\},$$4, the sequence is generalized quasi-semi-Jordan when deleting $$\sigma_\pi(A):=\{\lambda\in \sigma(A):|\lambda|=r(A)\},$$5 leaves a block that reverses about its center: $$\sigma_\pi(A):=\{\lambda\in \sigma(A):|\lambda|=r(A)\},$$6 When this holds, the generalized product is called quasi-semi-Jordan. The role of this symmetry is decisive: it is precisely the condition under which the adjoint-type alternative survives in the classification.

The proof strategy is organized around rank-one operators. A spectral-radius argument yields a rank-one characterization: $$\sigma_\pi(A):=\{\lambda\in \sigma(A):|\lambda|=r(A)\},$$7 has rank $$\sigma_\pi(A):=\{\lambda\in \sigma(A):|\lambda|=r(A)\},$$8 if and only if, for every $$\sigma_\pi(A):=\{\lambda\in \sigma(A):|\lambda|=r(A)\},$$9, the peripheral spectrum $r(A)$0 is a singleton. From the product-preservation identity, $r(A)$1 sends rank-one operators bijectively onto rank-one operators. A standard rank-one-injectivity argument then forces linearity and gives, on rank-one tensors $r(A)$2,

$r(A)$3

Linearity and the rank-one-generated ideals extend this to the full algebra, and comparison on reversed monomials detects the quasi-semi-Jordan symmetry in the adjoint case.

The Hilbert-space refinement treats skew generalized products

$r(A)$4

If $r(A)$5 preserves the peripheral spectrum of every such skew product and its range contains all rank-$r(A)$6 operators, then there exist a unitary or conjugate-unitary $r(A)$7 and a scalar $r(A)$8, with $r(A)$9 when $C^*$0 is odd, such that either

$C^*$1

for all $C^*$2, or, in the skew quasi-semi-Jordan case,

$C^*$3

Here $C^*$4 is the transpose in some fixed orthonormal basis. The proof passes through rank-one projections and then invokes the Uhlhorn–Wigner theorem after constructing a bijection on unit spheres that preserves $C^*$5.

3. Spectrum-preserving maps for power-type operator products

On an infinite-dimensional complex Hilbert space $C^*$6, Taghavi–Hosseinzadeh study surjective maps on $C^*$7 and $C^*$8 preserving the spectrum of specific products. The products include $C^*$9 with $A_{sa}\to B_{sa}$0 rational and $A_{sa}\to B_{sa}$1, $A_{sa}\to B_{sa}$2 with $A_{sa}\to B_{sa}$3 rational, and $A_{sa}\to B_{sa}$4 with $A_{sa}\to B_{sa}$5 rational and $A_{sa}\to B_{sa}$6. The corresponding theorems show that, in each case, a surjective map $A_{sa}\to B_{sa}$7 satisfying the prescribed spectral identity must have the form

$A_{sa}\to B_{sa}$8

for every $A_{sa}\to B_{sa}$9 in the domain, where $\sigma(\varphi(x)\varphi(y))=\sigma(xy)$0 is a bounded bijection that is either complex-linear or conjugate-linear and satisfies $\sigma(\varphi(x)\varphi(y))=\sigma(xy)$1 (Taghavi et al., 2013).

For self-adjoint operators, the first theorem assumes

$\sigma(\varphi(x)\varphi(y))=\sigma(xy)$2

for all $\sigma(\varphi(x)\varphi(y))=\sigma(xy)$3 and all $\sigma(\varphi(x)\varphi(y))=\sigma(xy)$4, and concludes that $\sigma(\varphi(x)\varphi(y))=\sigma(xy)$5 on $\sigma(\varphi(x)\varphi(y))=\sigma(xy)$6. The second theorem replaces $\sigma(\varphi(x)\varphi(y))=\sigma(xy)$7 by $\sigma(\varphi(x)\varphi(y))=\sigma(xy)$8 and reaches the same conclusion on $\sigma(\varphi(x)\varphi(y))=\sigma(xy)$9. For the full algebra $A^rBA^s$0, the third and fourth theorems impose the spectrum-preservation identities for all $A^rBA^s$1 and again conclude unitary or anti-unitary implementation.

The paper rephrases the conclusion as the statement that there is a unitary or anti-unitary $A^rBA^s$2 on $A^rBA^s$3 such that

$A^rBA^s$4

for all operators in the domain. In that sense, these power-type multiplicative spectrum preservers are exactly the Jordan-$A^rBA^s$5-automorphisms of $A^rBA^s$6 or $A^rBA^s$7.

The proofs follow a common pattern. First, one shows $A^rBA^s$8, $A^rBA^s$9, and preservation of rank-one projections in both directions. Next, functional-calculus arguments and results on rational powers imply preservation of rank-one operators and the trace pairings

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$00

for rank-one projections $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$01. These trace relations yield additivity and $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$02-homogeneity on $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$03, and then full complex-linearity on $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$04. A rank-one-preserver theorem then produces the conjugation formula. The operator-algebraic significance is that no a priori linearity or continuity is required beyond the stated spectral condition and surjectivity.

4. Self-adjoint parts of unital $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$05-algebras

Mori–Oi consider surjective maps

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$06

between the self-adjoint parts of unital $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$07-algebras satisfying

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$08

Their main theorem states that such a map is characterized by a central symmetry and a Jordan $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$09-isomorphism: there exist a central symmetry $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$10, meaning a self-adjoint unitary in the center of $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$11, and a Jordan $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$12-isomorphism $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$13 such that

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$14

Equivalently, $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$15 factors as $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$16 (Mori et al., 2024).

Two intermediate lemmas organize the proof. The real-spectrum test says that for $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$17, the condition $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$18 for every $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$19 is equivalent to

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$20

The uniqueness lemma says that if $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$21 satisfy

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$22

then $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$23. The proof also uses Jacobson’s lemma, square-root identities for positive elements, and an inductive spectral-inclusion argument.

The normalization step sets $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$24. Since $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$25, one gets $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$26, so $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$27 is a self-adjoint unitary. Surjectivity and the real-spectrum lemma then force $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$28 to be central. Defining

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$29

produces a unital surjection preserving the multiplicative spectrum on $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$30. On the cone $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$31 of invertible positive elements, the spectral condition shows that $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$32 is an isometry for the Thompson metric

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$33

A further application of the uniqueness lemma yields positive homogeneity: $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$34 for all $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$35. Hatori–Molnár’s theorem on Thompson-isometric positive-homogeneous surjections then gives a Jordan $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$36-isomorphism $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$37 extending $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$38.

In matrix algebras $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$39, the theorem specializes to four possibilities because every Jordan $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$40-automorphism of $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$41 is either $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$42 or $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$43, while every central symmetry is $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$44. Hence every surjective multiplicatively spectrum-preserving map on $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$45 is of one of the four forms

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$46

5. Stochastic matrices and their linear spans

For stochastic matrices, the multiplicative spectrum-preserving problem is formulated on $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$47 equal to the set of $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$48 doubly stochastic matrices $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$49, row stochastic matrices $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$50, column stochastic matrices $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$51, or the real linear span of one of these sets. A list of maps

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$52

is a multiplicative spectrum preserver if

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$53

for all $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$54. When $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$55, linearity is assumed. The same paper also treats multiplicative trace preservers and proves that when $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$56, multiplicative trace preservers always coincide with multiplicative spectrum preservers (Tsai et al., 26 Sep 2025).

A structural decomposition underlies the classification. With $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$57 the all-ones column and

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$58

one has

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$59

Thus every $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$60 decomposes uniquely as

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$61

Moreover, a suitable orthogonal similarity shows $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$62, and if $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$63 or $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$64, then

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$65

Hence both spectrum and trace of products depend only on the $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$66-parts.

On $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$67, the one-map case is classical in form: a linear $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$68 satisfies $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$69 for all $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$70 if and only if there exists $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$71 such that

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$72

For two maps $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$73, preservation of $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$74, equivalently $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$75, holds if and only if there are $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$76 with

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$77

or the transpose version

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$78

For $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$79, the classification becomes cyclic: there exist $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$80 with $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$81 such that

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$82

Exactly the same pattern holds on the span $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$83, except that the conjugating matrices $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$84 need only be invertible in the algebra $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$85. For $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$86 and $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$87,

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$88

For $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$89,

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$90

or the transpose version; in the special $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$91 case one must further insert a central invertible $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$92 on one side. For $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$93,

$$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$94

The $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$95 and $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$96 classifications are obtained by passing down from the $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$97-part. On $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$98, a linear one-map spectrum preserver agrees with a $$T_1 * T_2 * \cdots * T_k := T_{i_1}T_{i_2}\cdots T_{i_m},$$99-spectral automorphism on the $i_j\in\{1,\ldots,k\}$00-summand, while it is arbitrary on the $i_j\in\{1,\ldots,k\}$01-summand. For two maps, the $i_j\in\{1,\ldots,k\}$02-parts satisfy the two-map classification and the $i_j\in\{1,\ldots,k\}$03-parts are arbitrary maps into $i_j\in\{1,\ldots,k\}$04. On $i_j\in\{1,\ldots,k\}$05 itself, a linear spectrum preserver is, for $i_j\in\{1,\ldots,k\}$06, inner permutation-conjugation $i_j\in\{1,\ldots,k\}$07; for $i_j\in\{1,\ldots,k\}$08 it has the form

$i_j\in\{1,\ldots,k\}$09

For $i_j\in\{1,\ldots,k\}$10, the classification on $i_j\in\{1,\ldots,k\}$11 is

$i_j\in\{1,\ldots,k\}$12

where $i_j\in\{1,\ldots,k\}$13, $i_j\in\{1,\ldots,k\}$14, and $i_j\in\{1,\ldots,k\}$15 is any map such that the sum stays in $i_j\in\{1,\ldots,k\}$16. If one $i_j\in\{1,\ldots,k\}$17 is linear, then $i_j\in\{1,\ldots,k\}$18 is forced to be scalar on $i_j\in\{1,\ldots,k\}$19, and for $i_j\in\{1,\ldots,k\}$20 one obtains the rigid form $i_j\in\{1,\ldots,k\}$21.

6. Structural themes, proof mechanisms, and limitations

Across these settings, the classification theorems depend on a small number of recurring mechanisms. In the standard operator-algebra and Hilbert-space product problems, rank-one operators and rank-one projections are the decisive test objects; spectral preservation on carefully chosen products forces preservation of rank-one structure, after which linearity and inner implementation follow (Zhang et al., 2013, Taghavi et al., 2013). In the $i_j\in\{1,\ldots,k\}$22-algebraic setting, the key tools are spectral characterization on the self-adjoint part, normalization at the unit, the Thompson metric on invertible positives, and a passage from metric isometries to Jordan $i_j\in\{1,\ldots,k\}$23-isomorphisms (Mori et al., 2024). In the stochastic-matrix setting, the essential device is the decomposition into $i_j\in\{1,\ldots,k\}$24-, $i_j\in\{1,\ldots,k\}$25-, and $i_j\in\{1,\ldots,k\}$26-parts, combined with trace extremality of permutation matrices, the nondegenerate bilinear form $i_j\in\{1,\ldots,k\}$27, and Birkhoff’s theorem (Tsai et al., 26 Sep 2025).

Several limitations are explicit in the classifications. Adjoint-type or transpose-type alternatives occur only under symmetry hypotheses: the Banach-space anti-isomorphism case requires quasi-semi-Jordan symmetry, and the skew generalized product transpose case occurs only in the skew quasi-semi-Jordan case (Zhang et al., 2013). In stochastic matrices, the transpose map $i_j\in\{1,\ldots,k\}$28 on $i_j\in\{1,\ldots,k\}$29 cannot extend to a two-map preserver on $i_j\in\{1,\ldots,k\}$30, which illustrates that the direct-sum freedom in $i_j\in\{1,\ldots,k\}$31 is essential but constrained by the $i_j\in\{1,\ldots,k\}$32-nonnegativity requirement (Tsai et al., 26 Sep 2025). In the operator-product theorems on $i_j\in\{1,\ldots,k\}$33 and $i_j\in\{1,\ldots,k\}$34, surjectivity is part of every theorem statement, and the test sets may be restricted to $i_j\in\{1,\ldots,k\}$35 in the self-adjoint cases (Taghavi et al., 2013). In the peripheral-spectrum theorem, the range assumption that $i_j\in\{1,\ldots,k\}$36 contain all rank-$i_j\in\{1,\ldots,k\}$37 operators is likewise built into the result (Zhang et al., 2013).

Taken together, these results show that multiplicative spectrum preservation is not merely a weak spectral compatibility condition. Preserving only the peripheral spectrum of generalized products, preserving the full spectrum of self-adjoint products, or preserving product spectra on stochastic matrix sets all lead to forms dictated by the ambient algebraic symmetry. A plausible implication is that multiplicative spectral data often encode enough rank, order, or permutation structure to recover the underlying automorphism class, even when linearity is not assumed a priori.