Directed H-Toeplitz Graphs in Operator Algebras

• Directed H-Toeplitz Graphs are a family of related constructions arising from Fock space operator matrices that encode both Toeplitz and Hankel behaviors.
• They are characterized by graphical patterns such as diagonal bands and anti-diagonals, interpreted via relative Toeplitz graph algebras and hereditary data.
• Their study unifies operator theory, C*-algebra pullbacks, and generalized gauge actions to reveal deeper insights into graph structure and spectral reconstruction.

Directed H-Toeplitz graphs sit at the intersection of operator theory, directed graph $C^*$-algebras, and categorical graph constructions. In current literature the phrase is not uniform: it is introduced explicitly for digraphs extracted from the matrix of an $H$-Toeplitz operator on Fock space, while other papers state that the term does not appear explicitly in their text and instead interpret it through relative Toeplitz graph algebras, hereditary data, or generalized gauge actions by a compact abelian group $H$. This suggests that “Directed H-Toeplitz Graphs” is best understood as a family of related constructions rather than a single standardized definition (Singh et al., 9 Jul 2025, Brooker et al., 2022, Castro et al., 30 Mar 2025, Brownlowe et al., 2018).

1. Terminology and basic operator-algebraic framework

A directed graph is written $E=(E^0,E^1,r,s)$, with vertices $E^0$, edges $E^1$, range map $r:E^1\to E^0$, and source map $s:E^1\to E^0$. In the relative-graph setting, paths of length $n$ form $E^n$, $H$0 denotes the finite paths, and $H$1 the infinite paths. A vertex is singular if it is a source or an infinite receiver, and regular otherwise. Relative Toeplitz graph algebras are defined from a subset $H$2 by imposing the Cuntz–Krieger relation only on $H$3: $H$4 is universal for (CK1)–(CK3) together with

$H$5

Thus $H$6 and $H$7 (Brooker et al., 2022).

In the relation-morphism framework, the Toeplitz graph $H$8-algebra $H$9 of a directed graph $H$0 is the $H$1-enveloping algebra of the Cohn path algebra. It is universally generated by projections $H$2 and partial isometries $H$3 satisfying CK1 and the Toeplitz inequalities, including

$H$4

There is a canonical surjection $H$5 imposing equality at regular vertices (Castro et al., 30 Mar 2025).

A concise comparison of the principal usages is helpful.

Usage	Core object	Role of $H$6
Fock-space usage	Digraph from the matrix of $H$7	Part of “$H$8-Toeplitz operator”
Relative-graph usage	Relative graph $H$9, quotient ideals, pullbacks	Interpreted through hereditary data $E=(E^0,E^1,r,s)$0 or the Hajac framework
Reconstruction usage	Toeplitz algebra with generalized gauge action	Compact abelian group $E=(E^0,E^1,r,s)$1

The papers on relative graphs and relation morphisms explicitly state that the term “Directed H-Toeplitz Graphs” does not appear in their text. By contrast, the Fock-space paper introduces directed H-Toeplitz graphs as graphical encodings of operator matrices. The shared backbone across these usages is the Toeplitz-type passage from directed combinatorics to operator algebras (Singh et al., 9 Jul 2025, Brooker et al., 2022, Castro et al., 30 Mar 2025).

2. Fock-space directed H-Toeplitz graphs

In the operator-theoretic usage, the starting point is the Fock space $E=(E^0,E^1,r,s)$2 over $E=(E^0,E^1,r,s)$3 with Gaussian measure

$E=(E^0,E^1,r,s)$4

inner product

$E=(E^0,E^1,r,s)$5

orthonormal basis

$E=(E^0,E^1,r,s)$6

and reproducing kernel $E=(E^0,E^1,r,s)$7. The orthogonal projection $E=(E^0,E^1,r,s)$8 is

$E=(E^0,E^1,r,s)$9

For $E^0$0, one has $E^0$1 (Singh et al., 9 Jul 2025).

The symbol is harmonic,

$E^0$2

with

$E^0$3

The associated Toeplitz and Hankel operators are

$E^0$4

where $E^0$5. The dilation/flip operator $E^0$6 is defined on the basis by

$E^0$7

and the $E^0$8-Toeplitz operator is

$E^0$9

This unifies Toeplitz and Hankel parts: even-index basis vectors pass through the Toeplitz part, and odd-index basis vectors pass through the Hankel part (Singh et al., 9 Jul 2025).

With respect to $E^1$0, the matrix $E^1$1 of $E^1$2 is given by

$E^1$3

and

$E^1$4

Deleting every odd column yields the matrix of $E^1$5, while deleting every even column yields the matrix of $E^1$6 (Singh et al., 9 Jul 2025).

The directed H-Toeplitz graph attached to $E^1$7 has vertex set $E^1$8, indexing the basis $E^1$9. There is a directed edge $r:E^1\to E^0$0 if and only if $r:E^1\to E^0$1. One may weight the edge $r:E^1\to E^0$2 by $r:E^1\to E^0$3. The indegree and outdegree are

$r:E^1\to E^0$4

with weighted versions

$r:E^1\to E^0$5

In this sense the graph is a literal support digraph, or weighted support digraph, of the interlaced Toeplitz/Hankel matrix (Singh et al., 9 Jul 2025).

3. Diagonal bands, anti-diagonals, and graph structure

The matrix of $r:E^1\to E^0$6 produces two distinct adjacency geometries. Even columns are Toeplitz-like: for a fixed even column $r:E^1\to E^0$7, edges occur at $r:E^1\to E^0$8 whenever $r:E^1\to E^0$9, and at $s:E^1\to E^0$0 whenever $s:E^1\to E^0$1. Odd columns are Hankel-like: for a fixed odd column $s:E^1\to E^0$2, edges occur at all $s:E^1\to E^0$3 with $s:E^1\to E^0$4. Accordingly, even columns encode diagonal bands $s:E^1\to E^0$5, whereas odd columns encode anti-diagonal bands $s:E^1\to E^0$6 (Singh et al., 9 Jul 2025).

If $s:E^1\to E^0$7 and $s:E^1\to E^0$8 are polynomials of finite degrees $s:E^1\to E^0$9 and $n$0, then the Toeplitz portion has at most $n$1 upper diagonals and $n$2 lower diagonals with nonzero entries, while the Hankel portion has at most $n$3 anti-diagonals with nonzero entries, determined by indices $n$4 with $n$5. For finite polynomial symbols, the outdegree per column is uniformly bounded by the number of nonzero coefficients, and several examples show that the outdegree is often constant across columns for large indices. Indegree sequences can display parity dependence, reflecting both the arithmetic constraints in the binary abstraction $n$6 and the $n$7 indexing of the Hankel bands (Singh et al., 9 Jul 2025).

The paper introduces the combinatorial abstraction $n$8, with vertices $n$9 and an arc $E^n$0 if and only if

$E^n$1

for some $E^n$2 or $E^n$3. In examples, the sets $E^n$4 and $E^n$5 directly encode the upper- and lower-diagonal positions of nonzero entries in a binary adjacency matrix. This gives a finite or countable digraph model for the matrix support pattern at $E^n$6 (Singh et al., 9 Jul 2025).

Three explicit examples organize the qualitative behavior. For the purely anti-analytic symbol $E^n$7, odd columns vanish, the binary matrix has ones on three upper diagonals, and the digraph is encoded as $E^n$8. In this case $E^n$9 for all $H$00, while the indegrees begin

$H$01

For the purely analytic symbol $H$02, both Toeplitz and Hankel parts are present, the adjacency pattern is $H$03, and

$H$04

with indegrees beginning

$H$05

For the mixed symbol $H$06, the pattern is $H$07 and the outdegree is constantly $H$08 (Singh et al., 9 Jul 2025).

The same paper translates several operator-theoretic properties into graph-theoretic language. One has $H$09. For uniformly continuous harmonic $H$10, $H$11 is compact if and only if $H$12. For bounded harmonic $H$13, $H$14 is Hilbert–Schmidt if and only if $H$15; equivalently,

$H$16

Graphically, this means that nonzero H-Toeplitz operators yield weighted digraphs whose squared edge weights are not summable. For bounded analytic symbols satisfying the paper’s hypotheses, $H$17 if and only if $H$18 and $H$19 are linearly dependent; when that occurs, the corresponding directed H-Toeplitz graphs have identical adjacency patterns and proportional weights (Singh et al., 9 Jul 2025).

4. Relative graphs, hereditary data, and pullbacks of Toeplitz graph algebras

A second, substantially different, use of the phrase arises from relative graphs and relative Toeplitz graph algebras. The paper on relative graphs states that the term “Directed H-Toeplitz graphs” does not appear explicitly there, but interprets it through directed graphs equipped with relative Toeplitz algebras $H$20 and hereditary-relative quotient structure $H$21. In this setting a relative graph is a pair $H$22 with $H$23, and a morphism $H$24 is an injective graph homomorphism $H$25 such that $H$26 is hereditary in $H$27, $H$28, and $H$29 (Brooker et al., 2022).

Given a hereditary set $H$30, the associated subgraph $H$31 has

$H$32

The breaking vertices are

$H$33

and for $H$34 the gap projection is

$H$35

For $H$36, Theorem 3.5–Corollary 3.7 give a surjection

$H$37

if and only if $H$38, with kernel

$H$39

where

$H$40

Equivalently, $H$41 is the ideal generated by

$H$42

This extends the classical saturated-hereditary quotient $H$43 to the relative Toeplitz setting (Brooker et al., 2022).

The groupoid formulation clarifies the ideal theory. The graph groupoid $H$44 has unit space

$H$45

and $H$46. For $H$47, let $H$48 be the reduction of $H$49 to $H$50. Then

$H$51

and

$H$52

Gauge-invariant ideals of $H$53 are exactly those generated by $H$54 for open invariant $H$55, and such sets correspond bijectively to pairs $H$56 with $H$57 hereditary and $H$58 via

$H$59

The ideal corresponding to $H$60 is $H$61 (Brooker et al., 2022).

Pushouts exist in the category of relative graphs. Given morphisms $H$62, $H$63, the pushout $H$64 has graph part

$H$65

with the expected vertex and edge identifications, and relative set

$H$66

If $H$67, then $H$68, $H$69, and

$H$70

By Pedersen’s pullback criterion, the corresponding commuting square of quotient maps is a pullback if and only if the product of the ideals is zero (Brooker et al., 2022).

The main admissibility condition is

$H$71

Theorem 6.9 states that this is equivalent to several formulations: the outer square of relative Toeplitz algebras induced by the pushout is a pullback; $H$72; $H$73; and the decomposed singular/regular condition

$H$74

Under admissibility,

$H$75

This replaces the earlier prohibition on breaking vertices by a condition on where the Cuntz–Krieger relations are imposed (Brooker et al., 2022).

The examples isolate the obstruction. If $H$76 for all $H$77, the Toeplitz-algebra diagram is always a pullback. If $H$78, so all algebras are graph algebras, the outer diagram is a pullback if and only if

$H$79

A non-admissible example occurs when a vertex is breaking for both complements; then the outer graph-algebra square is not a pullback. An admissible example occurs when breaking happens in only one leg; then the graph-algebra square is a pullback (Brooker et al., 2022).

5. Functorial and reconstruction perspectives

A categorical treatment of Toeplitz graph $H$80-algebras is given by the relation category $H$81, whose morphisms are relations

$H$82

between finite-path sets. The subcategory $H$83 consists of multiplicative, decomposable, proper, partial-function-on-vertices relations that are also target bijective and monotone. For $H$84, the induced map on Cohn path algebras is

$H$85

and it lifts uniquely to a $H$86-homomorphism

$H$87

On generators,

$H$88

Regularity characterizes when the induced map factors from Toeplitz algebras through graph $H$89-algebras. The paper therefore places Toeplitz graph $H$90-algebras inside a contravariant functorial apparatus that unifies earlier covariant and contravariant constructions (Castro et al., 30 Mar 2025).

For finite directed graphs, a different role for $H$91 appears in generalized gauge actions. Here

$H$92

and the generalized gauge action

$H$93

is defined by

$H$94

The standard gauge action is the single-circle action $H$95, $H$96. The paper proves two reconstruction theorems: there exists a $H$97-isomorphism $H$98 intertwining the gauge actions and carrying $H$99 onto $H$00 if and only if $H$01; and if $H$02 and $H$03 have no sinks, then there exists an isomorphism $H$04 intertwining the generalized gauge actions $H$05 and $H$06 if and only if $H$07 (Brownlowe et al., 2018).

The reconstruction mechanism is spectral. For each vertex $H$08,

$H$09

satisfies

$H$10

With the $H$11-valued inner product $H$12, one has

$H$13

Hence the edge multiplicity from $H$14 to $H$15 is recovered as

$H$16

The same paper also uses KMS theory and the spectral subspaces for $H$17 to identify vertices and count paths, including the formula

$H$18

Thus $H$19 functions as a grading group that separates outgoing edges by source vertex (Brownlowe et al., 2018).

The limitations are equally sharp. The paper proves that the gauge action $H$20 alone is not enough: there exist non-isomorphic finite directed graphs $H$21 and $H$22 with a gauge-equivariant isomorphism

$H$23

and even the triples $H$24 and $H$25 can be isomorphic. The “no sinks” hypothesis is also necessary for reconstruction from $H$26 alone, because if $H$27 is a sink then the $H$28-coordinate of $H$29 is trivial on degree-one generators, so the source-wise spectral signature disappears (Brownlowe et al., 2018).

6. Higher-rank Toeplitz graphs and related extensions

A related but separate construction realizes Toeplitz algebras as Cuntz–Krieger algebras of enlarged graphs. For a row-finite $H$30-graph $H$31, the higher-rank graph $H$32 has vertices

$H$33

and morphisms $H$34 and $H$35 with

$H$36

$H$37

Composition is defined only by

$H$38

so there is no composition with $H$39 on the left. Consequently, paths ending at a $H$40-vertex are terminal. The graph $H$41 is row-finite and aperiodic (Pangalela, 2015).

If $H$42 is the universal Cuntz–Krieger family in $H$43, define

$H$44

Then $H$45 is a Toeplitz–Cuntz–Krieger $H$46-family, and the assignment $H$47 induces an isomorphism

$H$48

The key identity is that the Toeplitz inequality is realized as a Cuntz–Krieger equality plus an explicit remainder at the auxiliary $H$49-vertex: $H$50 Thus the $H$51-vertices record the gap between equality and inequality (Pangalela, 2015).

For $H$52, this specializes to a directed Toeplitz graph $H$53. The vertices are $H$54 for $H$55 and $H$56 whenever $H$57 emits an edge. For each original edge $H$58, there are edges

$H$59

There is no composition with $H$60 on the left, so $H$61-vertices are terminal. In $H$62,

$H$63

and the Toeplitz generators satisfy

$H$64

This gives a graph-theoretic dilation of the Toeplitz algebra into a genuine graph $H$65-algebra (Pangalela, 2015).

Taken together, these constructions show three distinct ways in which directed graphs become “H-Toeplitz.” They may be support digraphs of $H$66-Toeplitz operators on Fock space; relative graphs whose hereditary and quotient data govern pullbacks of relative Toeplitz graph algebras; or directed graphs encoded by Toeplitz algebras endowed with an $H$67-action, especially $H$68. A further implication, stated explicitly in the relative-graph paper, is that open directions include “working out pullbacks of relative Toeplitz algebras for higher-rank graphs using the general machinery for gauge-invariant ideals and groupoids,” which would connect the hereditary/pullback framework to the higher-rank Toeplitz graph $H$69 construction (Brooker et al., 2022, Pangalela, 2015).