Efficient Approaches for Verifying the Existence and Bound of Inverse of Linear Operators in Hilbert Spaces

Abstract

This paper describes some numerical verification procedures to prove the invertibility of a linear operator in Hilbert spaces and to compute a bound on the norm of its inverse. These approaches improve on previous procedures that use an orthogonal projection of the Hilbert space and its a priori error estimations. Several verified examples which confirm the effectiveness of the new procedures are presented.

1 Introduction

Let X, Y be complex Hilbert spaces endowed with inner products $(\,u,v\,)_{X}$, $(\,u,v\,)_{Y}$ and norms $\Vert u\Vert _X=\sqrt{(\,u,u\,)_{X}}$, $\Vert u\Vert _Y=\sqrt{(\,u,u\,)_{Y}}$, respectively, and $D({\mathcal {A}})$ be a complex Banach space satisfying $D({\mathcal {A}}) \subset X \subset Y$. We assume that the embedding $D({\mathcal {A}}) \hookrightarrow X$ is compact and that there exists an embedding constant $C_p >0$ for $X \hookrightarrow Y$ such that

$$\begin{aligned} \Vert u\Vert _Y \le C_p \Vert u\Vert _X, \quad \forall u \in X. \end{aligned}$$

(1)

Here, a concrete value of $C_p$ satisfying (1) should be estimated. We consider linear operators ${\mathcal {A}}: D({\mathcal {A}}) \rightarrow Y$, ${\mathcal {Q}}: X \rightarrow Y$, and

$$\begin{aligned} \mathscr {L}:= {\mathcal {A}}+ {\mathcal {Q}}: \quad D({\mathcal {A}}) \rightarrow Y. \end{aligned}$$

(2)

The aim of this paper is to present computer-assisted procedures for proving the invertibility of $\mathscr {L}$ defined by (2) and for computing a constant $M>0$ satisfying

$$\begin{aligned} \Vert \mathscr {L}^{-1} \phi \Vert _X \le M \Vert \phi \Vert _Y, \quad \forall \phi \in Y \end{aligned}$$

(3)

in a mathematically rigorous sense. The constant M in (3) represents an upper bound on the operator norm $\Vert \mathscr {L}^{-1}\Vert _{{{{\mathcal {L}}}}(Y,X)}$.

For example, when one aims at proving the existence of a solution of a nonlinear functional equation with a mathematically rigorous error bound by using Newton-type or Newton–Kantorovich-type arguments (e.g., [14, 17]), the operator $\mathscr {L}$ is the linearization of a given nonlinear problem. Therefore, the invertibility of $\mathscr {L}$ and bounding the norm for $\mathscr {L}^{-1}$ play essential roles in most computer-assisted rigorous approaches to nonlinear problems, and it is also desirable to obtain M in (3) as small as possible [7, 13,14,15, 17, 25]. We note that determining the existence and norm bounds for $\mathscr {L}^{-1}$ is also applied to eigenvalue exclosure for self-adjoint or non-self-adjoint eigenvalue problems in Hilbert spaces [11, 23]. Concerning nonlinear problems in Hilbert or Banach spaces, we are able to refer to powerful computer-assisted proofs for the ordinary/partial differential equations: Gameiro–Lessard [1], Arioli–Koch [2], Day–Lessard–Mischaikow [4], Figueras–Gameiro–Lessard–de la Llave [5], Nagatou–Plum–McKenna [12], Hungria–Lessard–Mireles James [8], van den Berg–Williams‘[3], Oishi [16], and references therein.

We have previously proposed two types of numerical verification approaches [10, 24] to assure the invertibility of $\mathscr {L}$ defined by (2) and to bound M in (3). These are based on orthogonal projections to Galerkin approximations with constructive a priori error estimations and can be applied to the case in which the operator $\mathscr {L}$ is non-self-adjoint.

The first approach in [10] transforms the problem $\mathscr {L}\psi = \phi$ for $\phi \in Y$ into an equivalent fixed-point problem on X, constructs a validated bound M, and verifies the invertibility of $\mathscr {L}$. Although numerous computer-assisted proofs have demonstrated the effectiveness of this approach [13, 14, 23], it has a restriction such that M in (3) does not converge to the exact operator norm $\Vert \mathscr {L}^{-1}\Vert _{{{{\mathcal {L}}}}(Y,X)}$ even though the dimension of the Galerkin space increases.

In [24], some of the authors considered another estimation for M in (3). This second approach avoided the fixed-point formulation and showed that M is expected to converge, as the the dimension of the Galerkin space increases, to its exact operator norm of $\mathscr {L}^{-1}$ under suitable assumptions. However, the invertibility criterion of the second approach in [24] is sometimes more difficult compared to that of the first approach, in which case it requires a greater computational cost [10, 23].

The aim of the present paper is to propose some improved procedures relative to those in [10, 24] obtaining (3), as well as new invertibility criteria for $\mathscr {L}$. The essential principle is, in fact, the same as that in [21] for second-order linear elliptic operators. However, the proposed procedures in this paper are a generalization of [21], and this generalization allows an extension to operators in Hilbert spaces other than second-order elliptic operators (e.g., see Sect. 5.2). These procedures are based on a projection and constructive a priori error estimations in Hilbert spaces as well as a computable upper bound $M_{{\mathcal {A}}}>0$ satisfying

$$\begin{aligned} \Vert {\mathcal {A}}u\Vert _Y \le M_{{\mathcal {A}}}\Vert \mathscr {L}u\Vert _Y, \quad \forall u \in D({\mathcal {A}}) \end{aligned}$$

(4)

which assures the invertibility of $\mathscr {L}$ under some assumptions (see the proof of Theorem 3). Although, in the ideal situation ($h \rightarrow 0$), our proposed approaches have the same bound M as in [10] or [24], some verification examples reveal that the proposed procedures have better bounds than the approach in [10] or [24]. We also note that the estimation (4) and the guaranteed upper bound $M_{{\mathcal {A}}}$ should be helpful for invertibility confirmation such as the eigenvalue excluding approach in [11] or the Newton–Kantorovich-type verification approach in [17].

Recently, Sekine–Nakao–Oishi [20] introduced the following $2 \times 2$ block operator matrix on $X_h \times X_*$ ($X=X_h \oplus X_*$, $\dim X_h < \infty$) representing $\mathscr {L}$, of the form

$$\begin{aligned} {{{\mathcal {D}}}} = \left( \begin{array}{cc} {{{\mathcal {T}}}} &{} {{{\mathcal {Y}}}} \\ {{{\mathcal {Z}}}} &{} {{{\mathcal {W}}}} \end{array} \right) : X_h \times X_* \rightarrow X_h \times X_* \end{aligned}$$

and proposed an interesting verification technique by effective use of ${{{\mathcal {D}}}}$. The method is based on the fact that, under the condition that the finite-dimensional operator ${{{\mathcal {T}}}}: X_h \rightarrow X_h$ is invertible, if its Schur complement defined by ${{{\mathcal {S}}}} = {{{\mathcal {W}}}} - {{{\mathcal {Z}}}}{{{\mathcal {T}}}}^{-1}{{{\mathcal {Y}}}}: X_* \rightarrow X_*$ is also invertible, then ${{{\mathcal {D}}}}$ itself is reversible. By using this principle, the solution of $\mathscr {L}\psi = \phi$ for $\phi \in Y$ can be enclosed without estimating the norm of $\mathscr {L}^{-1}$ directly, in a way similar to the Gaussian elimination method for simultaneous linear equations. In [20], they also presented a formulation of the numerical verification of the solution for nonlinear elliptic problems according to this methodology, which can be considered as an alternative approach to the method described in this paper. In the future, it will be important to compare the strengths and weaknesses of such verification methods with those described here.

The remainder of the present paper is organized as follows. The next section is devoted to the description of assumptions on the given linear operators and some finite-dimensional approximation subspaces with related constants. We summarize our previous results based on [10, 24] on the invertibility of $\mathscr {L}$ and M. Section 3 proposes two constructive norm estimations of (4), as well as the invertibility criteria of $\mathscr {L}$. In Sect. 4 we propose a new bound on the norm of the inverse of $\mathscr {L}$ under the invertibility condition described in the previous section and present various considerations. Several verification examples of the proposed procedures comparing them with the previous approaches are reported on in the final section.

2 Galerkin Approximation, Related Constants, and Known Results

This section describes assumptions on the linear operator $\mathscr {L}$ defined by (2) and introduces a finite-dimensional approximation subspace with related constants. Assume that the operator ${\mathcal {A}}$ has the following properties A1 and A2.

A1:

${\mathcal {A}}$ is bijective with bounded inverse $\mathscr {A}^{-1}: Y \rightarrow D({\mathcal {A}})$.

By using the embedding operator $I_{D({\mathcal {A}}) \hookrightarrow X}$, the operator ${\mathcal {A}}^{-1}:= I_{D({\mathcal {A}}) \hookrightarrow X} \mathscr {A}^{-1}: Y \rightarrow X$ is compact because of the compactness of the embedding $D({\mathcal {A}}) \hookrightarrow X$.

A2:

It holds that

$$\begin{aligned} (\,u,v\,)_{X} = (\,{\mathcal {A}}u,v\,)_{Y}, \quad \forall u \in D({\mathcal {A}}), \quad \forall v \in X. \end{aligned}$$

(5)

A typical example of A1 is when ${\mathcal {A}}$ is the Laplacian and A2 is derived from partial integration (see Sect. 5 or [14, Chapter 4]). From A1, A2, and (1), we have

$$\begin{aligned} \Vert {\mathcal {A}}^{-1} u\Vert _X \le C_p \Vert u\Vert _Y, \quad \forall u \in Y. \end{aligned}$$

(6)

We define a finite-dimensional approximation subspace $X_h \subset X$ dependent on the parameter $h>0$. For example, in the case of a partial differential equation problem, $X_h$ is a finite element subspace with mesh size h. We also define the orthogonal projection $P_h: X \rightarrow X_h$ and the projection $P_0: Y \rightarrow X_h$ by

$$\begin{aligned} (\,v-P_hv,v_h\,)_{X}= & {} 0, \quad \forall v_h \in X_h, \quad \forall v \in X, \end{aligned}$$

(7)

$$\begin{aligned} (\,v-P_0v,v_h\,)_{Y}= & {} 0, \quad \forall v_h \in X_h, \quad \forall v \in Y, \end{aligned}$$

(8)

respectively. Because $X_h$ is a closed subspace of X, each element $u \in X$ can be uniquely decomposed as $u = u_h + u_*$ for $u_h \in X_h$ and $u_* \in X_*:= (I-P_h)X$.

Now we assume that $P_h$, ${\mathcal {A}}$, and ${\mathcal {Q}}$ have the properties A3, A4, and A5.

A3:

There exists $C(h)>0$ satisfying $C(h)\rightarrow 0$ as $h \rightarrow 0$ and

$$\begin{aligned} \Vert (I - P_h)u\Vert _X \le C(h) \Vert {\mathcal {A}}u\Vert _Y, \quad \forall u \in D({\mathcal {A}}). \end{aligned}$$

(9)

A4:

There exist $\tau _1>0$ and $\tau _2>0$ such that

$$\begin{aligned} \Vert {\mathcal {Q}}u\Vert _Y \le \tau _1 \Vert P_h u\Vert _X + \tau _2\Vert (I-P_h)u\Vert _X, \quad \forall u \in X. \end{aligned}$$

(10)

A5:

There exists $\tau _3>0$ satisfying

$$\begin{aligned} \Vert P_h {\mathcal {A}}^{-1} {\mathcal {Q}}u_*\Vert _X \le \tau _3 \Vert u_*\Vert _X, \quad \forall u_* \in X_*. \end{aligned}$$

(11)

Assumption A3 corresponds to error estimation of the orthogonal projection $P_h$ defined by (7). We emphasize that the estimation (9) is indispensable in our argument, and the compactness of the embedding $D({\mathcal {A}}) \hookrightarrow X$ plays an essential role in obtaining the constant C(h) with the desired properties. Assumptions A4 and A5 indicate detailed information about the boundedness of the operator ${\mathcal {Q}}$. We note that concrete values of the constants C(h) and $\tau _i$ ($i=1,2,3$) have to be known and have to be evaluated rigorously, and the constants $\tau _i$ depend on ${\mathcal {Q}}$ and h. Concrete examples of C(h) and $\tau _i$ are shown in Sect. 5.

We define a basis function of $X_h$ by $\{\phi _i\}_{i=1}^N$ for $N := \dim X_h$ and $N \times N$ matrices G, $A_1$, $A_2$ by

$$\begin{aligned} G_{ij}&:= (\,{\phi }_j,{\phi }_i\,)_{X}+(\,{\mathcal {Q}}{\phi }_j,{\phi }_i\,)_{Y}, \end{aligned}$$

(12)

$$\begin{aligned} {[A_1]}_{ij}&:= (\,{\phi }_j,{\phi }_i\,)_{X}, \end{aligned}$$

(13)

$$\begin{aligned} {[A_2]}_{ij}&:= (\,{\phi }_j,{\phi }_i\,)_{Y}, \end{aligned}$$

(14)

for $1 \le i,j \le N$, respectively. Because $\phi _i$ ($i=1,\ldots ,N$) are the basis functions of $X_h \subset X \subset Y$ and $(\,\cdot ,\cdot \,)_{X}$ and $(\,\cdot ,\cdot \,)_{Y}$ are the inner products of X and Y, $A_i$ ($i=1,2$) is positive definite and can be decomposed as $A_i= L_i L_i^H$, where H denotes conjugate transposition. For example, $L_i$ is taken to be the Cholesky factor of $A_i$; i.e., $L_i$ is a lower triangular matrix. By using $L_i$, for $u_h \in X_h$ and ${\varvec{u}}=[u_i] \in \mathbb {C}^N$, it holds that

$$\begin{aligned} \Vert u_h\Vert _X=\sqrt{{\varvec{u}}^HA_1{\varvec{u}}}=\Vert L_1^H{\varvec{u}}\Vert _2, \quad \Vert u_h\Vert _Y=\sqrt{{\varvec{u}}^HA_2{\varvec{u}}}=\Vert L_2^H{\varvec{u}}\Vert _2. \end{aligned}$$

We also assume that G has an inverse and we define positive values $\rho$, ${\hat{\rho }}$ by

$$\begin{aligned} \rho:= & {} \Vert L_1^H G^{-1} L_1 \Vert _2, \end{aligned}$$

(15)

$$\begin{aligned} {\hat{\rho }}:= & {} \Vert L_1^H G^{-1} L_2 \Vert _2, \end{aligned}$$

(16)

respectively. In actual computation, $\rho$ and ${\hat{\rho }}$ are obtained as upper bounds of matrix 2-norms, and evaluation of them, including a proof of the invertibility of G, can be reduced to the verified computation of the maximum singular value of a matrix [19].

We next present two previous computer-assisted proofs of the invertibility of $\mathscr {L}$, as well as the computable bound of M in (3).

Theorem 1

[10, Theorem 3] If

$$\begin{aligned} \kappa := C(h) ( \rho \, \tau _1 \tau _3 + \tau _2) < 1, \end{aligned}$$

(17)

then $\mathscr {L}$ defined by (2) has an inverse and $M>0$ for (3) is obtained by

$$\begin{aligned} M = \dfrac{\sqrt{ \left( \rho ( C_p + C(h)(\tau _3 - C_p\tau _2) ) \right) ^2 + \left( C(h) (1 + \rho C_p \tau _1) \right) ^2 }}{1-\kappa }. \end{aligned}$$

(18)

Remark 1

Theorem 1 is based on a fixed-point problem on X and it is an improved version of that appearing in [23, Theorem 5.1]. In Theorem 1, assuming that $\kappa$ and C(h) converge to 0 as $h \rightarrow 0$ implies that M for (18) converges to $C_p \rho$. Generally, $M/(C_p \rho ) \rightarrow 1$ does not indicate convergence to the exact operator norm $\Vert \mathscr {L}^{-1}\Vert _{{{{\mathcal {L}}}}(Y,X)}$.

Theorem 2

[24, Theorem 1] If

$$\begin{aligned} {\hat{\kappa }} := C(h) \tau _2 \left( {\hat{\rho }} \tau _1 + 1 \right) < 1, \end{aligned}$$

(19)

then $\mathscr {L}$ defined by (2) has an inverse and $M>0$ for (3) is obtained by

$$\begin{aligned} M = \dfrac{\sqrt{{\hat{\rho }}^2 + C(h)^2(1+{\hat{\rho }} \tau _1)^2}}{1-{\hat{\kappa }}}. \end{aligned}$$

(20)

Remark 2

The matrix G is the Galerkin approximation of the operator $\mathscr {L}$ and ${\hat{\rho }}$ in (16) reflects an approximation of $\Vert \mathscr {L}^{-1}\Vert _{{{{\mathcal {L}}}}(Y,X)}$. In (20), ${\hat{\kappa }} \rightarrow 0$ and $C(h) \rightarrow 0$ as $h \rightarrow 0$ implies $M/{\hat{\rho }} \rightarrow 1$. Therefore, in Theorem 2, the estimation (20) is expected to converge to the exact operator norm of $\mathscr {L}^{-1}$, as $h \rightarrow 0$. However, it has been reported that sometimes the criterion ${\hat{\kappa }} < 1$ is harder to satisfy than $\kappa < 1$ for fixed h, experimentally [14].

3 Invertibility Verification of $\mathscr {L}$ and Constructive Norm Bounds for ${\mathcal {A}}$

In this section, we construct a computable upper bound $M_{{\mathcal {A}}}>0$ satisfying (4). The inequality (4) and A1 give that $\mathscr {L}$ is one-to-one. Then, by using the compactness of an operator ${\mathcal {A}}^{-1}{\mathcal {Q}}$ on X and the Fredholm alternative, the invertibility of $\mathscr {L}$ is assured (see the proof of Theorem 3, [10, proof of Theorem 3], [23, proof of Theorem 5.1]).

Let us define an Hermitian and positive semidefinite $N \times N$ matrix E by

$$\begin{aligned} {[}E]_{ij} = (\,{\mathcal {Q}}\phi _j,{\mathcal {Q}}\phi _i\,)_{Y}, \end{aligned}$$

(21)

and the related matrix 2-norms by

$$\begin{aligned} {\rho _{\texttt {L}_\texttt {1}}}&:= \sqrt{\left\| (G^{-1} L_1)^H E (G^{-1} L_1) \right\| _2 }, \end{aligned}$$

(22)

$$\begin{aligned} {\rho _{\texttt {L}_\texttt {2}}}&:= \sqrt{\left\| (G^{-1} L_2)^H E (G^{-1} L_2) \right\| _2 }, \end{aligned}$$

(23)

respectively. Note that if the matrix E given by (21) can be decomposed as $E=L_3 L_3^H$, then ${\rho _{\texttt {L}_\texttt {1}}}$ and ${\rho _{\texttt {L}_\texttt {2}}}$ can be written as

$$\begin{aligned} {\rho _{\texttt {L}_\texttt {1}}}= \left\| L_3^H G^{-1} L_1 \right\| _2, \quad {\rho _{\texttt {L}_\texttt {2}}}= \left\| L_3^H G^{-1} L_2 \right\| _2. \end{aligned}$$

We also note that the matrix E appears in the actual computations since it derives the quadrature form of estimation such that $\Vert {\mathcal {Q}}v_h\Vert _Y = \sqrt{{\varvec{v}}^H E {\varvec{v}}}$ for $v_h = \sum _{i=1}^N v_i \phi _i$ and ${\varvec{v}}=[v_i] \in \mathbb {C}^N$.

Using ${\rho _{\texttt {L}_\texttt {1}}}$, ${\rho _{\texttt {L}_\texttt {2}}}$, ${\hat{\rho }}$, and $\rho$, we have the following norm estimations.

Lemma 1

By representing each $u \in D({\mathcal {A}})$ as $u = u_h + u_*$ with $u_h = P_h u \in X_h$ and $u_* =(I-P_h)u \in X_*$, and by setting $f=\mathscr {L}u \in Y$, it holds that

$$\begin{aligned} \Vert {\mathcal {Q}}u_h\Vert _Y&\le {\rho _{\texttt {L}_\texttt {1}}}\Vert P_h {\mathcal {A}}^{-1} (f - {\mathcal {Q}}u_*)\Vert _X, \end{aligned}$$

(24)

$$\begin{aligned} \Vert {\mathcal {Q}}u_h\Vert _Y&\le {\rho _{\texttt {L}_\texttt {2}}}\Vert P_0 (f - {\mathcal {Q}}u_*)\Vert _Y, \end{aligned}$$

(25)

$$\begin{aligned} \Vert u_h\Vert _X&\le \rho \, \Vert P_h {\mathcal {A}}^{-1} (f - {\mathcal {Q}}u_*)\Vert _X, \end{aligned}$$

(26)

$$\begin{aligned} \Vert u_h\Vert _X&\le {\hat{\rho }} \, \Vert P_0 (f - {\mathcal {Q}}u_*)\Vert _Y. \end{aligned}$$

(27)

Proof

From (7), A2, and (8) we have

$$\begin{aligned} (\,u_h,v_h\,)_{X}&= (\,u,v_h\,)_{X} = (\,{\mathcal {A}}u,v_h\,)_{Y} = (\,f - {\mathcal {Q}}u,v_h\,)_{Y} \\&= -(\,{\mathcal {Q}}u_h,v_h\,)_{Y} + (\,f - {\mathcal {Q}}u_*,v_h\,)_{Y} \\&= -(\,{\mathcal {Q}}u_h,v_h\,)_{Y} + (\,P_0(f - {\mathcal {Q}}u_*),v_h\,)_{Y}, \quad \forall v_h \in X_h, \end{aligned}$$

then

$$\begin{aligned} (\,u_h,\phi _i\,)_{X} + (\,{\mathcal {Q}}u_h,\phi _i\,)_{Y} = (\,P_0(f - {\mathcal {Q}}u_*),\phi _i\,)_{Y}, \quad 1 \le i \le N. \end{aligned}$$

(28)

By setting

$$\begin{aligned} u_h = \sum _{i=1}^N u_i \phi _i, \quad {\varvec{u}} = [u_i] \in \mathbb {C}^N,\\ P_0 (f - {\mathcal {Q}}u_*) = \sum _{i=1}^N g_i \phi _i, \quad {\varvec{g}} = [g_i] \in \mathbb {C}^N, \end{aligned}$$

Equations (12), (14) and (28) give us ${\varvec{u}} = G^{-1} A_2 {\varvec{g}}$. Also, since the range of ${\mathcal {A}}^{-1}$ is $D({\mathcal {A}})$, the right-hand side of (28) is

$$\begin{aligned} (\,P_0(f - {\mathcal {Q}}u_*),\phi _i\,)_{Y}&= (\, {\mathcal {A}}{\mathcal {A}}^{-1}(f - {\mathcal {Q}}u_*),\phi _i\,)_{Y} \nonumber \\&=(\,{\mathcal {A}}^{-1}(f - {\mathcal {Q}}u_*),\phi _i\,)_{X} \nonumber \\&=(\,P_h{\mathcal {A}}^{-1}(f - {\mathcal {Q}}u_*),\phi _i\,)_{X} \end{aligned}$$

(29)

by A2 and (7); then by setting

$$\begin{aligned} P_h {\mathcal {A}}^{-1}(f - {\mathcal {Q}}u_*) = \sum _{i=1}^N v_i \phi _i, \quad {\varvec{v}} = [v_i] \in \mathbb {C}^N, \end{aligned}$$

Equations (12), (13) and (29) give us ${\varvec{u}} = G^{-1} A_1 {\varvec{v}}$. Therefore, setting $G^{-H}=(G^{-1})^H$, from (21) and (22), the estimation (24) holds by

$$\begin{aligned} \Vert {\mathcal {Q}}u_h\Vert _Y^2&= {\varvec{u}}^H E {\varvec{u}} = {\varvec{v}}^H A_1 G^{-H} E G^{-1} A_1 {\varvec{v}} \\&= (L_1^H {\varvec{v}})^H L_1^H G^{-H} E G^{-1} L_1 (L_1^H {\varvec{v}})\\&= (L_1^H {\varvec{v}})^H (G^{-1}L_1)^H E (G^{-1} L_1) (L_1^H {\varvec{v}})\\&\le \Vert (G^{-1}L_1)^H E (G^{-1} L_1)\Vert _2 \Vert L_1^H {\varvec{v}}\Vert _2^2 \\&\le ({\rho _{\texttt {L}_\texttt {1}}})^2 \Vert P_h {\mathcal {A}}^{-1}(f - {\mathcal {Q}}u_*)\Vert _X^2, \end{aligned}$$

and from (21) and (23), estimation (25) holds by

$$\begin{aligned} \Vert {\mathcal {Q}}u_h\Vert _Y^2&= {\varvec{u}}^H E {\varvec{u}} = {\varvec{g}}^H A_2 G^{-H} E G^{-1} A_2 {\varvec{g}} \\&= (L_2^H{\varvec{g}})^H L_2^H G^{-H} E G^{-1} L_2 (L_2^H {\varvec{g}})\\&= (L_2^H{\varvec{g}})^H (G^{-1}L_2)^H E (G^{-1} L_2) (L_2^H {\varvec{g}})\\&\le \Vert (G^{-1}L_2)^H E (G^{-1} L_2)\Vert _2 \Vert L_2^H{\varvec{g}}\Vert _2^2 \\&\le ({\rho _{\texttt {L}_\texttt {2}}})^2 \Vert P_0 (f - {\mathcal {Q}}u_*) \Vert _Y^2. \end{aligned}$$

Similarly, from (15), the estimation (26) is implied by

$$\begin{aligned} \Vert u_h\Vert _X&= \Vert L_1^H {\varvec{u}} \Vert _2 = \Vert L_1^H G^{-1} A_1 {\varvec{v}} \Vert _2 \\&= \Vert L_1^H G^{-1} L_1 L_1^H {\varvec{v}} \Vert _2 \\&\le \rho \Vert P_h {\mathcal {A}}^{-1}(f - {\mathcal {Q}}u_*) \Vert _X, \end{aligned}$$

and from (16), the estimation (27) is also obtained by

$$\begin{aligned} \Vert u_h\Vert _X&= \Vert L_1^H {\varvec{u}} \Vert _2 = \Vert L_1^H G^{-1} A_2 {\varvec{g}} \Vert _2 \\&= \Vert L_1^H G^{-1} L_2 L_2^H {\varvec{g}} \Vert _2 \\&\le \Vert L_1^H G^{-1} L_2\Vert _2 \Vert L_2^H {\varvec{g}} \Vert _2 \\&\le {\hat{\rho }} \Vert P_0 (f - {\mathcal {Q}}u_*) \Vert _Y. \end{aligned}$$

$\square$

By using Lemma 1, two invertibility criteria for $\mathscr {L}$ and constructive upper bounds satisfying (4) are obtained. The first is given below.

Theorem 3

If it holds that

$$\begin{aligned} {\kappa _{\texttt {L}_\texttt {1}}}:= C(h)({\rho _{\texttt {L}_\texttt {1}}}\tau _3 + \tau _2) < 1, \end{aligned}$$

(30)

then $\mathscr {L}: D({\mathcal {A}}) \rightarrow Y$ is invertible and bound $M_{{\mathcal {A}}}>0$ in (4) can be taken as

$$\begin{aligned} M_{{\mathcal {A}}}= \dfrac{1 + C_p{\rho _{\texttt {L}_\texttt {1}}}}{1-{\kappa _{\texttt {L}_\texttt {1}}}}. \end{aligned}$$

(31)

Proof

Since each $u \in D({\mathcal {A}})$ can be uniquely decomposed as $u=u_h+u_*$ with $u_h = P_hu$ and $u_*=(I-P_h)u$, setting $f = \mathscr {L}u$ we have

$$\begin{aligned} \Vert {\mathcal {A}}u\Vert _Y&= \Vert -{\mathcal {Q}}u_* -{\mathcal {Q}}u_h + f\Vert _Y \nonumber \\&\le \Vert {\mathcal {Q}}u_*\Vert _Y + \Vert {\mathcal {Q}}u_h\Vert _Y + \Vert f\Vert _Y . \end{aligned}$$

(32)

Using A3, A4, (24), (6), and A5, we have the following estimation:

$$\begin{aligned} \Vert {\mathcal {A}}u\Vert _Y&\le C(h) \tau _2 \Vert {\mathcal {A}}u\Vert _Y + {\rho _{\texttt {L}_\texttt {1}}}\Vert P_h {\mathcal {A}}^{-1} (f - {\mathcal {Q}}u_*)\Vert _X+ \Vert f\Vert _Y \\&\le C(h) \tau _2 \Vert {\mathcal {A}}u\Vert _Y + {\rho _{\texttt {L}_\texttt {1}}}\Vert {\mathcal {A}}^{-1} f\Vert _X + {\rho _{\texttt {L}_\texttt {1}}}\Vert P_h {\mathcal {A}}^{-1} {\mathcal {Q}}u_*\Vert _X + \Vert f\Vert _Y \\&\le C(h) \tau _2 \Vert {\mathcal {A}}u\Vert _Y + C_p {\rho _{\texttt {L}_\texttt {1}}}\Vert f\Vert _Y + {\rho _{\texttt {L}_\texttt {1}}}\tau _3 \Vert u_*\Vert _X + \Vert f\Vert _Y \\&\le C(h) ({\rho _{\texttt {L}_\texttt {1}}}\tau _3 + \tau _2) \Vert {\mathcal {A}}u\Vert _Y + ( 1+ C_p {\rho _{\texttt {L}_\texttt {1}}}) \Vert f\Vert _Y. \end{aligned}$$

Therefore, under the assumption ${\kappa _{\texttt {L}_\texttt {1}}}<1$, the estimation (4) holds with (31). In particular, (4) and A1 show that $\mathscr {L}$ is one-to-one. Furthermore, for any given $\phi \in Y$, the problem

$$\begin{aligned} \text {find\ \ } \psi \in D({\mathcal {A}}) \quad \text {such that} \quad \mathscr {L}\psi = \phi \end{aligned}$$

(33)

is equivalent to

$$\begin{aligned} \text {find } \psi \in X \quad \text {such that} \quad (I+{\mathcal {A}}^{-1}Q) \psi = {\mathcal {A}}^{-1} \phi . \end{aligned}$$

(34)

Since ${\mathcal {A}}^{-1}Q: X \rightarrow X$ is compact, the Fredholm alternative holds for problem (34), whereby $\mathscr {L}$ being one-to-one implies that (34), and hence (33), is uniquely solvable. Therefore, $\mathscr {L}$ is bijective and hence invertible. $\square$

We show the second approach for (4), which assures the invertibility of $\mathscr {L}$.

Theorem 4

If it holds that

$$\begin{aligned} {\kappa _{\texttt {L}_\texttt {2}}}:= C(h)\tau _2({\rho _{\texttt {L}_\texttt {2}}}+1) < 1, \end{aligned}$$

(35)

then $\mathscr {L}: D({\mathcal {A}}) \rightarrow Y$ is invertible and bound $M_{{\mathcal {A}}}>0$ in (4) is given by

$$\begin{aligned} M_{{\mathcal {A}}}= \dfrac{1 + {\rho _{\texttt {L}_\texttt {2}}}}{1-{\kappa _{\texttt {L}_\texttt {2}}}}. \end{aligned}$$

(36)

Proof

Since each $u \in D({\mathcal {A}})$ can be uniquely decomposed as $u=u_h+u_*$, where $u_h = P_hu$ and $u_*=(I-P_h)u$, by setting $f = \mathscr {L}u$, we obtain the same inequality (32) as in the case of Theorem 3. By using A3, A4, and (25), we get the following:

$$\begin{aligned} \Vert {\mathcal {A}}u\Vert _Y&\le C(h)\tau _2 \Vert {\mathcal {A}}u\Vert _Y + {\rho _{\texttt {L}_\texttt {2}}}\Vert P_0(f - {\mathcal {Q}}u_*)\Vert _Y + \Vert f\Vert _Y \\&\le C(h)\tau _2 \Vert {\mathcal {A}}u\Vert _Y + {\rho _{\texttt {L}_\texttt {2}}}\Vert {\mathcal {Q}}u_*\Vert _Y + (1+{\rho _{\texttt {L}_\texttt {2}}})\Vert f\Vert _Y \\&\le C(h)\tau _2 ( 1 + {\rho _{\texttt {L}_\texttt {2}}}) \Vert {\mathcal {A}}u\Vert _Y + (1+{\rho _{\texttt {L}_\texttt {2}}}) \Vert f\Vert _Y. \end{aligned}$$

Then under the assumption ${\kappa _{\texttt {L}_\texttt {2}}}<1$, the estimation (4) holds with (36).$\square$

Remark 3

The invertibility conditions $\{ \kappa , {\hat{\kappa }}, {\kappa _{\texttt {L}_\texttt {1}}}, {\kappa _{\texttt {L}_\texttt {2}}}\}<1$ for $\mathscr {L}$ depend on the values of $\rho \tau _1 \tau _3$, ${\hat{\rho }} \tau _1 \tau _2$, ${\rho _{\texttt {L}_\texttt {1}}}\tau _3$, and ${\rho _{\texttt {L}_\texttt {2}}}\tau _2$, respectively. The estimation (10) implies $\Vert {\mathcal {Q}}u_h\Vert _Y \le \tau _1 \Vert u_h\Vert _X$ for $u_h \in X_h$; therefore, from (24)–(27), it is expected that $\rho \tau _1 \tau _3 \ge {\rho _{\texttt {L}_\texttt {1}}}\tau _3$ and ${\hat{\rho }} \tau _1 \tau _2 \ge {\rho _{\texttt {L}_\texttt {2}}}\tau _2$ hold because the estimations of (24) and (25) could be more fine-grained and incorporate information about the operator ${\mathcal {Q}}$.

On the other hand, although it can be proved that ${\rho _{\texttt {L}_\texttt {2}}}\le {\rho _{\texttt {L}_\texttt {1}}}\, C_p$ by using the same procedure in [10, Lemma 1], the result of $\min \{{\rho _{\texttt {L}_\texttt {1}}}\tau _3, {\rho _{\texttt {L}_\texttt {2}}}\tau _2\}$ could depend on the given problem. In the ideal situation ($h \rightarrow 0$), the bound $M_{{\mathcal {A}}}>0$ in (4) is given by $1+C_p{\rho _{\texttt {L}_\texttt {1}}}$ by Theorem 3 and $1+{\rho _{\texttt {L}_\texttt {2}}}$ by Theorem 4. We present some comparisons in Sect. 5.

4 Upper Bound $M>0$

This section presents some formulas for upper bound M satisfying (3) which are expected to be an improvement on previous results (18) and (20).

Applying the invertibility theorems for $\mathscr {L}$ described in Sect. 3, we have the following theorem.

Theorem 5

Assume that there exists a constant $M_{{\mathcal {A}}}>0$ satisfying (4), as well as that $\mathscr {L}: D({\mathcal {A}}) \rightarrow Y$ has an inverse. Then upper bound M in (3) can be taken as

$$\begin{aligned} M = \sqrt{ \rho ^2 \left( C_p + s \tau _3 \right) ^2 + s^2 }, \end{aligned}$$

(37)

or

$$\begin{aligned} M = \sqrt{{\hat{\rho }}^2 \left( 1 + s \tau _2 \right) ^2 + s^2 }, \end{aligned}$$

(38)

for $s:=C(h)M_{{\mathcal {A}}}$.

Proof

For each $f \in Y$, we set $u = \mathscr {L}^{-1} f \in D({\mathcal {A}})$ and represent u as $u = u_h + u_*$ with $u_h = P_h u \in X_h$ and $u_* =(I-P_h)u \in X_*$. For the finite-dimensional term, inequalities (26), (6), and A5 imply that

$$\begin{aligned} \Vert u_h\Vert _X&\le \rho \Vert P_h {\mathcal {A}}^{-1}(f-{\mathcal {Q}}u_*)\Vert _X \nonumber \\&\le \rho \left( \Vert {\mathcal {A}}^{-1}f\Vert _X + \Vert P_h {\mathcal {A}}^{-1}{\mathcal {Q}}u_*\Vert _X \right) \nonumber \\&\le \rho \left( C_p \Vert f\Vert _Y + \tau _3 \Vert u_*\Vert _X \right) , \end{aligned}$$

(39)

and inequalities (27) and A4 imply that

$$\begin{aligned} \Vert u_h\Vert _X&\le {\hat{\rho }} \Vert P_0(f-{\mathcal {Q}}u_*)\Vert _Y \nonumber \\&\le {\hat{\rho }} \left( \Vert f\Vert _Y + \tau _2 \Vert u_*\Vert _X \right) . \end{aligned}$$

(40)

For the infinite-dimensional term, from A3 and (4), we have

$$\begin{aligned} \Vert u_*\Vert _X \le s \Vert f\Vert _Y. \end{aligned}$$

(41)

Therefore, by substituting (41) into (39) and (40), it follows that

$$\begin{aligned} \Vert u_h\Vert _X&\le \rho ( C_p + \tau _3 s ) \Vert f\Vert _Y, \end{aligned}$$

(42)

$$\begin{aligned} \Vert u_h\Vert _X&\le {\hat{\rho }} ( 1 + \tau _2 s ) \Vert f\Vert _Y, \end{aligned}$$

(43)

respectively, and then $\Vert u\Vert _X^2 = \Vert u_h\Vert _X^2 + \Vert u_*\Vert _X^2$ implies the desired conclusions. $\square$

Remark 4

When $h \rightarrow 0$, the estimation in (37) shows that $M/(\rho \, C_p) \rightarrow 1$ and the estimation in (38) shows that $M/{\hat{\rho }} \rightarrow 1$. Therefore, in the ideal situation ($h \rightarrow 0$), we have the same bound M, namely, (37) versus (18) and (38) versus (20).

Remark 5

In order to obtain bound $M_{{\mathcal {A}}}>0$ in (4) by Theorems 3 or 4, the additional computational cost of the 2-norm of ${\rho _{\texttt {L}_\texttt {1}}}$ or ${\rho _{\texttt {L}_\texttt {2}}}$ is required. According to our numerical examples in the next section, the additional costs for ${\rho _{\texttt {L}_\texttt {1}}}$ or ${\rho _{\texttt {L}_\texttt {2}}}$ could be seen as approximately the same order of magnitude for $\rho$ or ${\hat{\rho }}$.

5 Verification Examples

This section reports on several computer-assisted results for the estimations of M in (3) proposed by Theorems 1, 2 and 5, as well as the invertibility verification of $\mathscr {L}$.

All computations were carried out on the Fujitsu PRIMERGY CX2570 M4; Intel Xeon Gold 6140 (Skylake-SP); 2.3 GHz (Turbo 3.7 GHz) by using INTerval LABoratory Version 11, a toolbox in MATLAB R2019a (9.7.0.1261785) 64-bit (glnxa64) developed by Rump [18] for self-validating algorithms. Therefore, all numerical values in these tables are verified data in the sense of mathematically strict rounding error control.

5.1 Second-order Differential Operators

For a unit square region $\Omega =(0,1)\times (0,1)$ and some integer m, let $H^{m}(\Omega )$ denote the complex $L^2$-Sobolev space of order m on $\Omega$. We define the Hilbert space $H^1_0(\Omega ):= \{ u \in H^{1}(\Omega ) \ | \ u=0 \ \text {on} \ \partial \Omega \}$ with inner product $(\,\nabla u,\nabla v\,)_{L^2(\Omega )}$ and norm $\Vert u\Vert _{H^1_0(\Omega )}:=\Vert \nabla u\Vert _{L^2(\Omega )}$, where $(\,u,v\,)_{L^2(\Omega )}$ denotes the $L^2$-inner product on $\Omega$. We consider a two-dimensional second-order non-self-adjoint operator

$$\begin{aligned} \mathscr {L}u = -\Delta u + b\cdot \nabla u + c u: \quad H^{2}(\Omega )\cap H^1_0(\Omega )\rightarrow L^2(\Omega )\end{aligned}$$

(44)

for $b \in L^\infty (\Omega )^2$ and $c \in L^\infty (\Omega )$ with norms

$$\begin{aligned} \Vert b\Vert _{L^\infty (\Omega )^2}= {\displaystyle \displaystyle \textrm{ess} \sup \limits _{x \in \Omega }}\sqrt{|b_1(x)|^2 + |b_2(x)|^2}, \quad \Vert c\Vert _{L^\infty (\Omega )}={\displaystyle \displaystyle \textrm{ess} \sup \limits _{x \in \Omega }}|c(x)|, \end{aligned}$$

respectively. We are able to set

$$\begin{aligned} D({\mathcal {A}})= & {} H^{2}(\Omega ) \cap H^1_0(\Omega ), \quad X = H^1_0(\Omega ), \quad Y = L^2(\Omega ), \quad {\mathcal {A}}= -\Delta , \nonumber \\ {\mathcal {Q}}= & {} b \cdot \nabla + c , \quad (\,u,v\,)_{X} = (\nabla u,\nabla v)_{L^2(\Omega )}, \quad (\,u,v\,)_{Y} = (u,v)_{L^2(\Omega )}. \end{aligned}$$

(45)

It is well known that A1 holds [6], and A2 is obtained by partial integration. We form a linear and uniform triangular mesh on $\Omega$ with the side length of each finite element being $h>0$ and set $X_h$ as a classical P1 finite element subspace of $H^1_0(\Omega )$. Then, for A3, $P_h$ is now the usual $H^1_0$-projection, and it is well known that (7) holds with $C(h)=0.493h$ and $C_p=1/(\pi \sqrt{2})$ [14].

Concerning A4 and A5, we can take

$$\begin{aligned} \tau _1 = \Vert b\Vert _{L^\infty (\Omega )^2} \ + C_p \Vert c\Vert _{L^\infty (\Omega )}, \ \tau _2 = \Vert b\Vert _{L^\infty (\Omega )^2} \ + C(h) \Vert c\Vert _{L^\infty (\Omega )}, \ \tau _3 = C_p \tau _2. \end{aligned}$$

Note that when b is differentiable, we can derive more accurate estimates for $\tau _i$ $(i=1,2,3)$ [21]. For more general setting of the second-order operator see, e.g., [14].

5.1.1 Example 1

Our first verification example is a non-self-adjoint case of

$$\begin{aligned} b(x_1,x_2) = R \begin{bmatrix}-x_2+0.5 \\ x_1-0.5 \end{bmatrix}, \quad c \in \mathbb {C}, \quad R>0 \end{aligned}$$

in (44), which comes from a stationary convection-diffusion equation. Tables 1 and 2 show the verification results of the invertibility and bound M for $(R,c)=(10,-1-1.5i)$ and for $(R,c)=(1,100+50i)$, respectively. In the tables herein, values in bold are the smallest M among (37), (38), (18), and (20), except for Table 3, and the symbol “–” indicates that the invertibility criterion does not hold. The value $M_{{\mathcal {A}}}$ adopted is the smaller of the results from (31) and (36). The computational cost is concentrated in the singular value computations; elapsed times for obtaining $\rho$, ${\hat{\rho }}$, ${\rho _{\texttt {L}_\texttt {2}}}$, and ${\rho _{\texttt {L}_\texttt {1}}}$ are 19.08, 16.05, 17.12 and 17.44 s, respectively, for $1/h=40$. In Table 1, (37) gives the best result. This may be because the value of $C_p \rho$ is close to ${\hat{\rho }}$ which is the approximation of the exact operator norm $\Vert \mathscr {L}^{-1}\Vert _{{{{\mathcal {L}}}}(L^2,H^1_0)}$. On the other hand, the difference between (37) and (38) tends to get smaller as the value of 1/h increases.

Table 1 Verification results of Example 1 for $R=10$ and $c=-1-1.5i$

Table 2 Verification results of Example 1 for $R=1$ and $c=100+50i$

5.1.2 Example 2

The next example is the case of a linearized equation of semilinear PDEs:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = 2(1 + u + u^2) -u^3/50 &{}\quad \text {in}\quad (0,1)\times (0,1), \\ u = 0 &{}\quad \text {on}\quad \partial \Omega . \end{array}\right. \end{aligned}$$

(46)

Equation (46) comes from a reaction-diffusion equation and, by the Newton–Raphson method with usual floating-point arithmetic, we have (at least) three finite element approximate solutions denoted by $u_h$. We refer to these approximate solution as “1st,” “2nd,” and “3rd” Fig. 1 shows the shape of $u_h$.

Fig. 1

Approximate solutions of (46)

The linearized operator $\mathscr {L}$ at $u_h$ is defined by

$$\begin{aligned} \mathscr {L}u = -\Delta u + (- 2 - 4u_h +3u_h^2/50)u, \end{aligned}$$

(47)

which is the case of $b = 0$. Tables 3, 4 and 5 present the verification results of the linearized operator defined by (47).

Table 3 Verification results of Example 2 (1st)

Table 4 Verification results of Example 2 (2nd)

Table 5 Verification results of Example 2 for (3rd)

For the case of the 1st solution, Eqs. (37), (38), (18) and (20) give almost the same bound M. For the case of the 2nd and 3rd solutions, the effectiveness of our proposed procedure and the validity of Theorem 5 are shown. It seems that (38) gives the smallest bound of M. The computational cost is concentrated in the singular value computations; elapsed times for obtaining $\rho$, ${\hat{\rho }}$, ${\rho _{\texttt {L}_\texttt {2}}}$, and ${\rho _{\texttt {L}_\texttt {1}}}$ are 5.81, 3.82, 4.29 and 4.46 s, respectively, for $1/h=40$.

5.2 Fourth-order Differential Operators

Consider the following fourth-order non-self-adjoint differential operator:

$$\begin{aligned} \mathscr {L}u = (-D^2+a^2)^2 u + iaRe[ V(-D^2+a^2)+V'']u - \mu (-D^2+a^2)u, \end{aligned}$$

(48)

which maps from $H^4(\Omega )\cap H^2_0(\Omega )$ to $L^2(\Omega )$. Here,

$$\begin{aligned} H^2_0(\Omega ) := \left\{ \, v \in H^2(\Omega ) \ | \ v(-1)=v'(-1)=v(1)=v'(1)=0 \, \right\} , \end{aligned}$$

for $\Omega =(-1,1)$, $V=1-x^2$, and the symbols $D=d/dx$, i, $a>0$, and $Re>0$ denote respectively the differential operator, the imaginary unit, the wave number, and the Reynolds number. The operator (48) comes from the well-known Orr–Sommerfeld equation, which is one of the central equations governing the linearized stability theory of the plane Poiseuille flow. For more details and applications, see, e.g., [10, Section 5.2].

For the operator (48), we can take $D({\mathcal {A}}) = H^4(\Omega )\cap H^2_0(\Omega )$, $X = H^2_0(\Omega )$, $Y = L^2(\Omega )$, with inner products

$$\begin{aligned} (\,u,v\,)_{X} = ((-D^2+a^2) u, (-D^2+a^2)v)_{L^2(\Omega )}, \quad (\,u,v\,)_{Y} = (u,v)_{L^2(\Omega )}, \end{aligned}$$

and ${\mathcal {A}}= (-D^2+a^2)^2$, ${\mathcal {Q}}= iaRe[ V(-D^2+a^2)+V''] -\mu (-D^2+a^2)$, $C_p = 1/(\pi ^2/4+a^2)$, respectively. We introduce a finite-dimensional approximation subspace $X_h \subset H^2_0(\Omega )$, using base functions constructed from piecewise cubic Hermite interpolating polynomials with uniform partition size h. Then we can take

$$\begin{aligned} C(h) =\frac{\sqrt{3}}{p} h^2 \left( 1 + \frac{a^2}{p}h^2\right) , \ \nu _1 = C(h) s_1, \ \nu _2 = s_2 + \tau _3 C_p, \ \nu _3 = s_2 + C(h) s_3 , \end{aligned}$$

where $s_1 := 2 s_3 \, C_p+ s_2 + Re \Vert V'\Vert _{L^\infty (\Omega )}$, $s_2:= \Vert -iaReV + \mu \Vert _{L^\infty (\Omega )}$, $s_3:= aRe\Vert V''\Vert _{L^\infty (\Omega )}$, and $p = 6\sqrt{70}/\sqrt{4+\sqrt{5}}$ [9, 22, 26].

5.2.1 Example 3

We take $Re=5776$ and $a=1.019$. Tables 6, 7 and 8 show the verification results of the operator (48) for $\mu = -100+1552.59i$, $\mu = -200+1552.59i$, and $\mu = -500+1552.59i$, respectively. Let us remark that, especially for h smaller than 1/400, the round-off error for $\rho$ and ${\rho _{\texttt {L}_\texttt {1}}}$ tends to be a serious problem (see also [23]). For each case, the effectiveness of our proposed procedure and the validity of Theorem 5 are also shown. In both Tables 6, 7 and 8, it can be seen that, as 1/h increases, (38) gives the best results and tends to approach the value of $\rho$, which is an approximation of the operator norm $\Vert \mathscr {L}^{-1}\Vert _{{{{\mathcal {L}}}}(L^2,H^2_0)}$. The computational cost is concentrated in the singular value computations; elapsed times for obtaining $\rho$, ${\hat{\rho }}$, ${\rho _{\texttt {L}_\texttt {2}}}$, and ${\rho _{\texttt {L}_\texttt {1}}}$ are 5.39, 4.62, 4.72 and 4.66 s, respectively, for $1/h=500$.

Table 6 Verification results of Example 3 for $\mu =-100+1552.59i$

Table 7 Verification results of Example 3 for $\mu =-200+1552.59i$

Table 8 Verification results of Example 3 for $\mu =-500+1552.59i$

Remark 6

In the verification examples treated in the present paper, we focus on linear operators only. We emphasize again that also in the context of computer-assisted proofs for nonlinear equations including ordinary/partial differential equations, the verification of the invertibility of $\mathscr {L}$ and the computation of a norm bound on $\mathscr {L}^{-1}$ plays an essential role, and our proposed approaches are expected to provide accurate and efficient enclosure results for solutions of nonlinear problems. We will report on concrete computer-assisted proofs of nonlinear equations using our proposed method in subsequent papers.

6 Conclusion

We proposed some improvements on the computer-assisted procedure for verifying the invertibility of a linear operator in a Hilbert space and the computation of a norm bound of its inverse. Although an additional computation of a 2-norm bound is required, verification examples confirm that the upper bound given by (37) improves on (18) and the upper bound given by (38) improves on (20) except for the case shown in Table 3.

The invertibility criteria in Theorems 3 and 4 and the guaranteed upper bound $M_{{\mathcal {A}}}$ should be helpful for invertibility confirmation such as eigenvalue exclosure for self-adjoint or non-self-adjoint eigenvalue problems in Hilbert spaces in [11, 23]. We conclude that an appropriate procedure can be selected taking into consideration the given problem or the computational cost, for example.