Dual Truncated Toeplitz Operator (DTTO)

• Dual truncated Toeplitz operators (DTTOs) are bounded linear operators defined on the complement of a model space, characterized by block matrix forms that blend Toeplitz and Hankel components.
• They employ a dual analytic/coanalytic factorization of unimodular symbols to determine norm-attainment, spectral properties, and Fredholm indices using classical operator theory.
• DTTOs serve as central models in modern functional analysis, enabling advances in nearly invariant subspace structure, block operator equivalence, and multicomponent Toeplitz systems.

A dual truncated Toeplitz operator (DTTO) is a bounded linear operator associated with a symbol $\varphi \in L^{\infty}(\mathbb{T})$, acting on the orthogonal complement ${\mathcal{K}_u}^{\perp} = uH^{2} \oplus H^{2}_{-}$ of the model space $\mathcal{K}_u = H^{2}\ominus uH^{2}$ for a fixed nonconstant inner function $u$. The DTTO is defined analogously to the classical truncated Toeplitz operator (TTO), but operates on the complementary subspace, leading to distinct functional-analytic and algebraic properties. DTTOs are central objects in recent work on model space theory, nearly invariant subspaces, multicomponent Toeplitz structures, operator-algebraic questions such as Fredholmness, and the interplay between analytic and coanalytic factorization phenomena (Bhuia et al., 14 Jan 2026, O'Loughlin, 2020, Câmara et al., 2020, Câmara et al., 2019, Wang et al., 2020).

1. Functional and Operator-Theoretic Setup

Let $H^{2}$ denote the Hardy space on the unit circle $\mathbb{T}$, with $H^{2}_{-} = L^{2} \ominus H^{2}$ the backward Hardy space. For a nonconstant inner function $u$, the model space is $\mathcal{K}_u = H^{2} \ominus uH^{2}$, and its orthogonal complement in $L^{2}$ is given by

$\mathcal{K}_u^{\perp} = uH^{2} \oplus H^{2}_{-}.$

Given a symbol $\varphi \in L^{\infty}(\mathbb{T})$, the DTTO $D_\varphi$ is defined on $h \in \mathcal{K}_u^{\perp}$ as

$D_\varphi(h) = (I - P_{\mathcal{K}_u})[\varphi h],$

where $P_{\mathcal{K}_u}$ denotes the orthogonal projection onto $\mathcal{K}_u$. Alternatively, using projective identities, $D_\varphi$ can be represented as

$$D_\varphi(h) = (I - P_+)[\varphi h] + uP_+[\bar{u} \varphi h],$$

with $P_+$ the Riesz projection onto $H^2$, resulting in a natural separation into analytic ($uH^2$) and coanalytic ($H^2_{-}$) components.

DTTOs admit a block operator matrix form under a unitary identification between $uH^{2} \oplus H^{2}_{-}$ and $H^{2} \oplus H^{2}_{-}$: $$U^* D_\varphi U = \begin{bmatrix} T_\varphi & H^*_{u\bar\varphi} \ H_{u\varphi} & S_\varphi \end{bmatrix}$$ where $T_\varphi$ is the classical Toeplitz operator, $S_\varphi$ the dual Toeplitz on $H^2_{-}$, and $H_{u\varphi}$ the Hankel operator with symbol $u\varphi$ (Bhuia et al., 14 Jan 2026, Wang et al., 2020, Câmara et al., 2019).

2. Norm, Boundedness, and Block-Matrix Realizations

For $\varphi \in L^{\infty}(\mathbb{T})$, $D_\varphi$ is bounded and

$\|D_\varphi\| = \|\varphi\|_\infty.$

The block-matrix realization of DTTOs allows direct application of the machinery of matrix-valued Toeplitz and Hankel operators. This block structure underpins equivalence after extension with paired or block Toeplitz operators, making available the full range of Fredholm and spectral theory from classical Toeplitz setting (Bhuia et al., 14 Jan 2026, Câmara et al., 2020). In the more general situation, DTTOs can be embedded into block (even $4 \times 4$) Toeplitz matrices, facilitating analysis via Wiener-Hopf and Riemann-Hilbert techniques, especially for dual-band variants (Câmara et al., 2020).

3. Analytic/Coanalytic Dichotomy and Norm-Attainment

A core structural dichotomy governs when $D_\varphi$ attains its norm (i.e., there exists $x \neq 0$ with $\|D_\varphi x\| = \|D_\varphi\| \|x\|$). For unimodular symbols with $\|\varphi\|_\infty=1$, norm-attainment occurs if and only if $\varphi$ admits one of two canonical inner factorizations:

• Analytic case:

$$\varphi = \bar{u} \overline{\psi_+} \chi_+ \qquad (\psi_+, \chi_+ \text{ inner}),$$

yielding extremal vectors lying in $uH^2$.

• Coanalytic case:

$$\varphi = u \psi_- \overline{\chi_-} \qquad (\psi_-, \chi_- \text{ inner}),$$

with extremals in $H^2_{-}$.

These cases are mutually exclusive and collectively exhaustive (Bhuia et al., 14 Jan 2026). In the analytic case, the extremal subspace is $$E_\varphi^{(+)} = \{u \psi_+ u_1 h \oplus 0 : h \in H^2\}$$ for $u$, $\chi_+$ decomposed via their greatest common divisors; the coanalytic extremal subspace in the respective case is $$E_\varphi^{(-)} = \{0 \oplus \overline{\psi_-} \overline{u_1} k : k \in H^2_-\}$$.

4. Kernel Structure and Nearly Invariant Subspaces

The kernel of a DTTO exhibits a fine structure closely related to nearly backward shift-invariant subspaces of Hardy and vector-valued Hardy spaces (O'Loughlin, 2020):

• For an inner function $u$ and symbol $g \in L^\infty$, $\ker D_g^u$ is always a finite-dimensional extension of a truly invariant subspace.
• There are isometric identifications of $g \ker D_g^u$ as a vector-valued shift-invariant subspace of $H^2(\mathbb{C}^2)$, with at most defect two.
• For invertible $g$, $\ker D_g^u = (g^{-1}u)K_{z\psi}$, relating the DTTO kernel to classical TTO kerneled subspaces (Hayashi-Hitt theory).

The decomposition theorem asserts that $\ker D_g^u$ is generated by up to two orthonormal basis elements with the entire subspace modeled via an invariant subspace of $H^2$ or $H^2(\mathbb{C}^2)$, depending on the defect (O'Loughlin, 2020).

5. Fredholmness, Invertibility, and Equivalence after Extension

DTTOs with invertible symbol $\varphi \in GL^\infty(\mathbb{T})$ are Fredholm with index zero, and invertibility is determined by invertibility of an associated classical or asymmetric TTO, under explicit isomorphisms (Câmara et al., 2019). More generally, DTTOs are equivalent after extension to paired matrix operators on $L^2 \oplus L^2$ of the form $AP_+ + BP_-$, with explicit 2x2 matrix symbols $A$, $B$. Fredholmness is characterized by invertibility of both $\det A = -u\varphi$ and $\det B = \varphi^{-1}$, yielding a Fredholm index of zero when satisfied.

In the dual-band generalization, compressions to $\varphi K_u \oplus \psi K_u$ and their operators $T^M_g$ are equivalent after extension to block Toeplitz operators with $4 \times 4$ matrix symbol, with Fredholm and spectral properties entirely inherited from classical Toeplitz theory under this identification (Câmara et al., 2020).

6. Spectral and Commutator Theory

DTTO spectra are determined by the essential range of the symbol $\varphi$:

• For continuous $\varphi$, $$\sigma_{\text{ess}}(D_{u,u}(\varphi)) = \varphi(\mathbb{T})$$.
• For $\varphi \in H^\infty$, the spectrum coincides with the closure of $\varphi(\mathbb{D})$ (Câmara et al., 2019).

Commutator and semi-commutator theory for DTTOs is fully characterized in terms of localization on support sets of the Douglas algebra $H^\infty + C$, generalizing the Brown–Halmos commutativity criterion. Specifically, two DTTOs $D_f$, $D_g$ essentially commute if and only if, for every support set, their symbols satisfy one of three mutually exclusive analytic, coanalytic, or trivial linear-combination conditions. Similar structural dichotomies hold for the compactness of semi-commutators (Wang et al., 2020). These results, employing block operator decompositions and function algebra techniques, extend classical essential-commutativity theory into the model space-complement setting of DTTOs.

7. Concrete Examples and Applications

• For $u(z) = z$ and $\varphi(z) = z^{-k}, |k|\geq2$, the analytic norm-attainment case is realized; the extremal vector is $f = z \oplus z^{-1}$.
• For $\varphi(z) = (1+z)/2$ (with $|\varphi|<1$ on a set of positive measure), $D_\varphi$ does not attain its norm.
• The dual compressed shift $D_u$ (i.e., $\varphi(z) = z$ and $u(z) = z$) always attains its norm and has the closed unit disk as spectrum.

Applications of DTTO theory include explicit computation of nearly invariant subspace structure, operator-theoretic realization of Hardy space invariance phenomena, and block operator model theory for multi-band convolution and function theory (Bhuia et al., 14 Jan 2026, O'Loughlin, 2020, Câmara et al., 2020).

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DTTOs thus encode a deep, operator-theoretic reflection of the analytic/coanalytic structure of their symbols, provide gateways between function-theoretic, algebraic, and operator-theoretic analysis, and serve as central models for a range of problems in Hardy space theory, spectral theory, and the modern study of function-theoretic operator compressions.