Nearly Invariant Subspaces: Theory & Applications

• Nearly invariant subspaces are closed subspaces of reproducing kernel Hilbert spaces that satisfy a weakened invariance condition, such as f(0)=0 implying f/z belongs to the space.
• They are typically represented in the Hardy space as gK_I, where the extremal function g carries the geometry and isometric properties from the model space K_I.
• Extensions of the theory include finite defect analogues, applications to truncated Toeplitz operators, de Branges spaces, semigroups, and multivariable operator models.

Nearly invariant subspaces are closed subspaces that satisfy a weakened invariance condition under a backward shift, a division operator, or an adjoint semigroup action. In the classical Hardy-space setting, a closed subspace $M\subset H^2$ is nearly invariant if

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$

Hitt’s theorem gives the canonical structure

$M=gK_I,\qquad K_I=H^2\ominus IH^2,$

where $I$ is inner with $I(0)=0$, $g$ is the extremal function of $M$, and multiplication by $g$ is an isometric isomorphism from $K_I$ onto $M$. Much of the subsequent theory consists in determining how far this model-space paradigm extends, how it interacts with Toeplitz-type operators and boundary values, and how it changes when one allows finite defect, passes to other reproducing kernel Hilbert spaces, or replaces the backward shift by semigroups, products of shifts, or finite-rank perturbations (Hartmann et al., 2011).

1. Classical Hardy-space structure

In $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$0, nearly invariant subspaces are organized by the model-space representation $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$1. The extremal function $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$2 is the unique solution of

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$3

and the map

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$4

is an isometric isomorphism. Sarason’s characterization of isometric multipliers states that every such $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$5 has the form

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$6

where $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$7 lie in the unit ball and satisfy

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$8

This formula guarantees that $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$9 is a closed subspace of $M=gK_I,\qquad K_I=H^2\ominus IH^2,$0 (Hartmann et al., 2011).

The reproducing-kernel geometry of $M=gK_I,\qquad K_I=H^2\ominus IH^2,$1 is inherited from $M=gK_I,\qquad K_I=H^2\ominus IH^2,$2 through $M=gK_I,\qquad K_I=H^2\ominus IH^2,$3. Since

$M=gK_I,\qquad K_I=H^2\ominus IH^2,$4

the orthogonal projection onto $M=gK_I,\qquad K_I=H^2\ominus IH^2,$5 is

$M=gK_I,\qquad K_I=H^2\ominus IH^2,$6

and the reproducing kernel of $M=gK_I,\qquad K_I=H^2\ominus IH^2,$7 is

$M=gK_I,\qquad K_I=H^2\ominus IH^2,$8

Accordingly,

$M=gK_I,\qquad K_I=H^2\ominus IH^2,$9

The kernel of a nearly invariant subspace is therefore the model-space kernel weighted by the extremal multiplier (Hartmann et al., 2011).

This classical picture remains the reference point for much of the subject. In later work on compressed shifts, nearly $I$0-invariant subspaces continue to be written in the form $I$1, where $I$2 is the extremal function from Hitt’s theorem and $I$3; the difference is that operator-theoretic questions are then transported from $I$4 to $I$5 through the unitary multiplier $I$6 (Liang et al., 23 Jun 2025).

2. Truncated Toeplitz operators and boundary values

On a nearly invariant subspace $I$7, truncated Toeplitz operators are defined by

$I$8

whenever this makes sense. The central intertwining identity is

$I$9

where the right-hand side is the truncated Toeplitz operator on $I(0)=0$0. Consequently,

$I(0)=0$1

From this, the standard structural properties of truncated Toeplitz operators on model spaces pass to nearly invariant subspaces: $I(0)=0$2 is weakly closed; $I(0)=0$3 iff $I(0)=0$4; $I(0)=0$5 is a family of complex symmetric operators with respect to

$I(0)=0$6

and a bounded operator $I(0)=0$7 on $I(0)=0$8 belongs to $I(0)=0$9 iff there are $g$0 such that

$g$1

where $g$2. Rank-one truncated Toeplitz operators are likewise transported from $g$3, and the nontrivial selfadjoint rank-one case is described by the boundary kernels $g$4 for $g$5 (Hartmann et al., 2011).

Boundary regularity is more delicate than the algebraic structure. For model spaces, the Ahern–Clark criterion says that every $g$6 has a finite non-tangential limit at $g$7 iff $g$8, equivalently iff the kernels $g$9 are uniformly bounded in every Stolz region at $M$0. For nearly invariant spaces $M$1, every function in $M$2 has a finite non-tangential limit at $M$3 iff two conditions hold: $M$4 itself has a finite non-tangential limit at $M$5, and

$M$6

for every Stolz region $M$7. The paper emphasizes that kernel boundedness alone is not enough; there are examples in which the kernels are uniformly bounded but $M$8 has no boundary limit, so not every function in $M$9 has a boundary limit there. A corresponding dichotomy then holds: if every function in $g$0 has a non-tangential limit at $g$1, either $g$2, or $g$3, where every function in $g$4 tends to $g$5 non-tangentially at $g$6 (Hartmann et al., 2011).

The compressed shift on a nearly invariant space exhibits the same transport principle. For $g$7,

$g$8

is unitarily equivalent to a rank-one perturbation $g$9 of the classical compressed shift, where

$K_I$0

Using the Frostman shift

$K_I$1

and the Crofoot transform, one obtains

$K_I$2

$K_I$3

and

$K_I$4

This gives a complete spectral and invariant-subspace classification for compressed shifts on nearly $K_I$5-invariant subspaces (Liang et al., 23 Jun 2025).

3. Finite defect and almost invariance

A major enlargement of the theory replaces exact near invariance by near invariance with finite defect. In the scalar Hardy space, $K_I$6 is nearly $K_I$7-invariant with defect $K_I$8 if there exists an $K_I$9-dimensional subspace $M$0, usually taken orthogonal to $M$1, such that

$M$2

The basic representation theorem states that if $M$3 contains a function not vanishing at $M$4, then

$M$5

where $M$6 is the normalized reproducing kernel of $M$7 at $M$8, $M$9 is an orthonormal basis of the defect space, $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$00 is a closed subspace of a vector-valued Hardy space invariant under a direct sum of backward shifts, and

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$01

If every function in $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$02 vanishes at $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$03, only the defect part remains. This theorem is a finite-defect Beurling–Hitt–Sarason description, and the converse also holds (Chalendar et al., 2019).

The same pattern extends to vector-valued Hardy spaces. For $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$04, a nearly $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$05-invariant subspace with defect $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$06 has the form

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$07

or, in the vanishing-at-zero case,

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$08

where $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$09 is $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$10-invariant and the norm splits orthogonally. These results were then used to describe almost invariant subspaces for $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$11 and $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$12, to connect scalar and vector-valued nearly invariant subspaces, and to model kernels of Toeplitz-type operators through backward-shift-invariant coefficient spaces (Chattopadhyay et al., 2020, O'Loughlin, 2020).

A parallel literature studies almost invariant subspaces in the operator-theoretic sense

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$13

with $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$14 finite-dimensional. In this branch, finite-rank perturbations are central: $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$15 is almost invariant for $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$16 iff $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$17 is invariant for $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$18 for some finite-rank $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$19. For the backward shift and its perturbations, invariant subspaces of operators such as

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$20

yield almost invariant or nearly $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$21-invariant subspaces, and conversely almost invariant subspaces arise from suitable finite-rank perturbations. One explicit representation is

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$22

with $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$23 a vector-valued model space. A later reformulation expresses $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$24-almost invariant subspaces as ranges

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$25

and proves the striking equivalence

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$26

on vector-valued Hardy spaces (Das et al., 2024, Gu et al., 2024).

In Banach-space operator theory, almost-invariant subspaces are also studied independently of Hardy-space division. There the formal definition is again

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$27

with $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$28 finite-dimensional, and the defect is the smallest possible $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$29. Every bounded operator on an infinite-dimensional separable reflexive Banach space admits an almost-invariant half-space with defect one, equivalently a rank-one perturbation with an invariant half-space. Related results show that triangularizable quasinilpotent operators, triangularizable operators with countable spectrum on reflexive spaces, and polynomially compact operators admit almost-invariant half-spaces; in a Hilbert-space MASA setting, the finite-rank commutator condition $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$30 of finite rank for every projection $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$31 forces a decomposition $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$32 with $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$33 in the MASA and $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$34 finite rank (Liang et al., 2020, Marcoux et al., 2012).

4. Other analytic Hilbert spaces

In de Branges spaces, near invariance is formulated as a zero-division property: if $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$35 vanishes at $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$36, then $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$37, at least away from common zeros. For a nearly invariant subspace $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$38 with no common zeros, the structure theorem is completely rigid: $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$39 for some de Branges space $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$40 and some $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$41. The proof analyzes the reproducing kernel

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$42

derives $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$43, shows that

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$44

must be an entire zero-free Nevanlinna-class function with unimodular boundary values on $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$45, hence

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$46

and then renormalizes $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$47 into a genuine de Branges space. In the Paley–Wiener case this yields the characterization

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$48

for an interval $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$49 (Malman, 2019).

Weighted Fock-type spaces display a different rigidity threshold. In spaces $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$50 of finite order with dense polynomials, every nontrivial backward shift invariant subspace is

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$51

the polynomials of degree at most $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$52; in the radial case the same conclusion holds for $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$53. For nearly invariant subspaces, an analogue of de Branges’ Ordering Theorem holds in the zero exponential type regime: the family of nearly invariant subspaces is totally ordered by inclusion, and if polynomials are dense then every nontrivial nearly invariant subspace is again some $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$54. This fails in larger-growth Fock-type spaces, where nontrivial infinite-dimensional nearly invariant subspaces exist and ordering breaks down (Aleman et al., 2020).

For abstract shift operators of finite multiplicity, and for multiplication by a finite Blaschke product $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$55 on Dirichlet-type spaces $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$56, near invariance is expressed as

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$57

or, with finite defect,

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$58

Such subspaces are modeled by vector-valued Hardy-space backward-shift invariant spaces. In the finite-multiplicity shift case,

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$59

while in the Dirichlet-type setting the general form is

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$60

with $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$61 invariant under an appropriate backward shift associated with $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$62, possibly after rescaling when $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$63 (Liang et al., 2020, Chattopadhyay et al., 2020).

Further generalization occurs for Hilbert spaces contractively contained in reproducing kernel Hilbert spaces. If $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$64 is nearly invariant under division by an inner function $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$65 and satisfies the norm monotonicity condition used in the paper, then $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$66 is represented by a multiplier matrix $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$67 acting on an $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$68-invariant space $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$69,

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$70

with only the inequality

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$71

for $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$72 in general. A finite-defect version adds a defect component carried by $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$73. The paper emphasizes that, beyond the Hardy-space case, neither isometricity nor closedness of $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$74 should be expected (Khan et al., 2023).

The real Hardy space $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$75 admits a direct real analogue of Hitt’s theorem. A nonzero nearly $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$76-invariant subspace has the form

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$77

where $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$78 is $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$79-invariant in $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$80, $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$81 is orthogonal to $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$82, $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$83, and multiplication by $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$84 is isometric on $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$85. Finite-defect versions reproduce the complex Chalendar–Gallardo-Gutiérrez–Partington model in the real setting, and yield a characterization of almost invariant subspaces for the real backward shift (Khan et al., 11 Apr 2026).

5. Semigroups, automorphisms, and multivariable extensions

For $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$86-semigroups, the naive analogue of backward-shift near invariance is vacuous, so the condition is reformulated as follows: a subspace $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$87 is nearly $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$88-invariant if

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$89

For the shift semigroup on $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$90,

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$91

minimal cyclic nearly invariant subspaces generated by delayed exponentials and their polynomially weighted variants can be computed explicitly. For example,

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$92

and, under the Laplace transform and the disk model,

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$93

More generally, closures of spaces $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$94 can be described as finite-codimensional subspaces of larger model spaces when $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$95 has the rational form specified in the paper (Liang et al., 2020).

Discrete semigroups generated by automorphisms of the disk preserve nearly invariant structure through composition operators. If $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$96 is an automorphism, then

$$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$97

and a Hitt-type theorem holds: $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$98 By contrast, if $$f\in M,\ f(0)=0 \quad\Longrightarrow\quad \frac{f}{z}\in M.$$99 is an inner function that is not an automorphism and $M=gK_I,\qquad K_I=H^2\ominus IH^2,$00, then $M=gK_I,\qquad K_I=H^2\ominus IH^2,$01 is not nearly $M=gK_I,\qquad K_I=H^2\ominus IH^2,$02-invariant. In particular, $M=gK_I,\qquad K_I=H^2\ominus IH^2,$03 is nearly $M=gK_I,\qquad K_I=H^2\ominus IH^2,$04-invariant iff $M=gK_I,\qquad K_I=H^2\ominus IH^2,$05 is an automorphism, assuming $M=gK_I,\qquad K_I=H^2\ominus IH^2,$06 (Liang et al., 2023).

In several variables, the one-variable definition must be modified. On $M=gK_I,\qquad K_I=H^2\ominus IH^2,$07, the coordinate-wise near-invariance condition is too strong for Toeplitz kernels, so the appropriate definition uses the product shift

$M=gK_I,\qquad K_I=H^2\ominus IH^2,$08

and

$M=gK_I,\qquad K_I=H^2\ominus IH^2,$09

A closed subspace $M=gK_I,\qquad K_I=H^2\ominus IH^2,$10 is then nearly $M=gK_I,\qquad K_I=H^2\ominus IH^2,$11-invariant if

$M=gK_I,\qquad K_I=H^2\ominus IH^2,$12

With this definition, kernels of Toeplitz operators on the bidisk are nearly invariant, and a bidisk version of the vector-valued Hitt theorem holds: $M=gK_I,\qquad K_I=H^2\ominus IH^2,$13 with $M=gK_I,\qquad K_I=H^2\ominus IH^2,$14 and $M=gK_I,\qquad K_I=H^2\ominus IH^2,$15 isometric on the model space. The same paper extends the construction to commuting pure isometric tuples, where kernels of general Toeplitz operators are nearly invariant for the product isometry (Zhu et al., 2024).

A different higher-order generalization concerns simultaneous near invariance for non-cyclic shift semigroups. For $M=gK_I,\qquad K_I=H^2\ominus IH^2,$16, $M=gK_I,\qquad K_I=H^2\ominus IH^2,$17, and $M=gK_I,\qquad K_I=H^2\ominus IH^2,$18, closed subspaces nearly invariant under both $M=gK_I,\qquad K_I=H^2\ominus IH^2,$19 and $M=gK_I,\qquad K_I=H^2\ominus IH^2,$20 are characterized by representations

$M=gK_I,\qquad K_I=H^2\ominus IH^2,$21

where $M=gK_I,\qquad K_I=H^2\ominus IH^2,$22 is built from orthogonalized reproducing kernels and the inner matrix $M=gK_I,\qquad K_I=H^2\ominus IH^2,$23 satisfies

$M=gK_I,\qquad K_I=H^2\ominus IH^2,$24

The corresponding invariant theory for $M=gK_I,\qquad K_I=H^2\ominus IH^2,$25 and $M=gK_I,\qquad K_I=H^2\ominus IH^2,$26 uses the same matrix condition, and the entire framework transfers to multiplication by finite Blaschke products through unitary equivalence (Liang et al., 2024).

6. Toeplitz kernels and contemporary operator models

Toeplitz kernels are among the most persistent sources of nearly invariant subspaces. In the classical Hardy space, $M=gK_I,\qquad K_I=H^2\ominus IH^2,$27 is nearly $M=gK_I,\qquad K_I=H^2\ominus IH^2,$28-invariant. For finite-rank perturbations

$M=gK_I,\qquad K_I=H^2\ominus IH^2,$29

the kernels $M=gK_I,\qquad K_I=H^2\ominus IH^2,$30 are nearly $M=gK_I,\qquad K_I=H^2\ominus IH^2,$31-invariant with finite defect. The defect space can be made explicit in several cases: when $M=gK_I,\qquad K_I=H^2\ominus IH^2,$32, one may take

$M=gK_I,\qquad K_I=H^2\ominus IH^2,$33

when $M=gK_I,\qquad K_I=H^2\ominus IH^2,$34 is inner,

$M=gK_I,\qquad K_I=H^2\ominus IH^2,$35

when $M=gK_I,\qquad K_I=H^2\ominus IH^2,$36 with $M=gK_I,\qquad K_I=H^2\ominus IH^2,$37,

$M=gK_I,\qquad K_I=H^2\ominus IH^2,$38

and for $M=gK_I,\qquad K_I=H^2\ominus IH^2,$39 with $M=gK_I,\qquad K_I=H^2\ominus IH^2,$40 inner, the defect space involves both $M=gK_I,\qquad K_I=H^2\ominus IH^2,$41 and the projections $M=gK_I,\qquad K_I=H^2\ominus IH^2,$42. The Chalendar–Gallardo–Partington theorem then represents these perturbed kernels through backward-shift-invariant subspaces in vector-valued Hardy spaces (Liang et al., 2019).

The same organizing principle governs kernels of truncated Toeplitz and related operators. A vector-valued finite-defect theory shows that kernels of scalar truncated Toeplitz operators, multiband truncated Toeplitz operators, and dual truncated Toeplitz operators fit into a common nearly invariant framework. In particular, the kernel of a multiband truncated Toeplitz operator is nearly $M=gK_I,\qquad K_I=H^2\ominus IH^2,$43-invariant with defect $M=gK_I,\qquad K_I=H^2\ominus IH^2,$44, and this bound is sharp. The general method is to realize the kernel as an isometric image of a backward-shift-invariant space, possibly after passing to a matrix-valued Toeplitz kernel (O'Loughlin, 2020).

Recent work emphasizes that nearly invariant subspaces are not merely auxiliary constructions but stable operator models in their own right. Invariant subspaces of finite-rank perturbations of the backward shift, Toeplitz–Hankel range descriptions, generalized nearly $M=gK_I,\qquad K_I=H^2\ominus IH^2,$45-invariance, semigroup versions, and compressed-shift models all retain a common architecture: a distinguished wandering or extremal part, a backward-shift-invariant coefficient space, and, when present, a finite-dimensional defect correction. A plausible implication is that the subject is best viewed not as a single theorem about $M=gK_I,\qquad K_I=H^2\ominus IH^2,$46, but as a family of representation theories for subspaces that are stable under division or adjoint action up to a controlled error term (Das et al., 2024, Gu et al., 2024, Liang et al., 23 Jun 2025).

Within this broader framework, a common misconception is that “nearly invariant” always means the Hardy-space condition $M=gK_I,\qquad K_I=H^2\ominus IH^2,$47. The literature supports a more differentiated view. In Hardy-space and model-space theory, that division property is the standard definition; in de Branges spaces the relevant division is by $M=gK_I,\qquad K_I=H^2\ominus IH^2,$48; in semigroup theory the condition is $M=gK_I,\qquad K_I=H^2\ominus IH^2,$49 for some $M=gK_I,\qquad K_I=H^2\ominus IH^2,$50; and in Banach-space operator theory the formal term is often “almost invariant,” meaning invariance modulo a finite-dimensional defect. What remains constant across these settings is the interplay between reproducing kernels, backward-shift or left-inverse structure, and finite-rank correction phenomena (Malman, 2019, Liang et al., 2020, Liang et al., 2020)