Pseudo Numerical Ranges and Spectral Enclosures

We introduce the new concepts of pseudo numerical range for operator functions and families of sesquilinear forms as well as the pseudo block numerical range for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\!\times \! n$$\end{document}n×n operator matrix functions. While these notions are new even in the bounded case, we cover operator polynomials with unbounded coefficients, unbounded holomorphic form families of type (a) and associated operator families of type (B). Our main results include spectral inclusion properties of pseudo numerical ranges and pseudo block numerical ranges. For diagonally dominant and off-diagonally dominant operator matrices they allow us to prove spectral enclosures in terms of the pseudo numerical ranges of Schur complements that no longer require dominance order 0 and not even \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<\!1$$\end{document}<1. As an application, we establish a new type of spectral bounds for linearly damped wave equations with possibly unbounded and/or singular damping.


Introduction
Spectral problems depending non-linearly on the eigenvalue parameter arise frequently in applications, see e.g.the comprehensive collection in [2] or the monograph [21]. The dependence ranges from quadratic in problems originating in second order Cauchy problems such as damped wave equations, see e.g. [13,15], to rational as in electromagnetic problems with frequency dependent materials such as photonic crystals, see e.g. [9], [1]. In addition, if energy dissipation is present due to damping or lossy materials, then the values of the corresponding operator functions need not be selfadjoint.
While for operator functions T (λ), λ ∈ ⊆ C, with unbounded operator values in a Hilbert space H the notion of numerical range W (T ) exists, hence they coincide with the so-called ε-pseudo numerical range first considered in [10]. As a consequence, the pseudo numerical range W (T ) can equivalently be described as W (T ) = λ ∈ : 0 ∈ W (T (λ)) =:W ,0 (T ). (1.2) One could be tempted to think that the condition 0 ∈ W (T (λ)) in W ,0 (T ) is equivalent to λ / ∈ W (T ), but this is neither true for operator functions with bounded values, as already noted in [31], nor for non-monic linear operator pencils for which the set W ,0 (T ) was used recently in [3].
One of the crucial properties of the pseudo numerical range is that, without any assumptions on the operator family, σ ap (T ) ⊆ W (T ), see Theorem 3.1, and that the norm of the resolvent of T can be estimated by

(T )\W ε (T ) ⊆ ρ(T )\W (T ).
Not only from the analytical point of view, but also from a computational perspective, the pseudo numerical range seems to be more convenient since it is much easier to determine whether a number is small rather than zero.
Like the numerical range of an operator function, but in contrast to the numerical range or essential numerical range of an operator [4,12,17], the pseudo numerical range need not be convex. An exception is the trivial case of a monic linear operator pencil T (λ) = A−λI , λ ∈ C, where the pseudo numerical range is simply the closure of the numerical range, W (T ) = W (T ) = W (A). In general, we only have the obvious enclosure W (T ) ⊆ W (T ). Neither the interiors nor the closures in of W (T ) and W (T ) need to coincide and there is also no inclusion either way between W (T ) or its closure W (T ) ∩ in and the closure W (T ) ∩ of W (T ) in ; we give various counter-examples to illustrate these effects.
In our first main result we use the pseudo numerical range of holomorphic form families t(λ), λ ∈ , of type (a) to prove the spectral inclusion for the associated holomorphic operator functions T (λ), λ ∈ , of type (B) of m-sectorial operators T (λ). More precisely, we show that if there exist k ∈ N 0 , μ ∈ and a core D of t(μ) with , λ ∈ C, which is a weaker condition than m-sectoriality of all T (λ).
The second new concept we introduce in this paper is the pseudo block numerical range of operator functions L(λ), λ ∈ , that possess an operator matrix representation with respect to a decomposition H = H 1 ⊕ · · · ⊕ H n , n ∈ N, of the given Hilbert space H. This means that with operator functions L i j (λ), λ ∈ , of densely defined and closable linear operators from H j to H i , i, j = 1, . . . , n.
Extending earlier concepts we first define the block numerical range of L as for bounded values L(λ) see [23] and [28] for n = 2, for unbounded operator matrices L(λ) = A − λI H see [24]. Then we introduce the pseudo block numerical range of L as For n = 1 both block numerical range and pseudo block numerical range coincide with the numerical range and pseudo numerical range of L, respectively. For n > 1, the trivial inclusion W n (L) ⊆ W n (L) and the characterisation (1.1), i.e. W n (L) = λ ∈ : 0 ∈ W n (L(λ)) , n ∈ N, and a resolvent norm estimate see Theorem 4.10 for both, continue to hold, but otherwise not much carries over from the case n = 1. The first difference is that, for the simplest case L(λ) = A − λI H , λ ∈ C, we may have W n (L) = W n (L) for n > 1, see Example 4.5. More importantly, for n > 1 the relation (1.2) need not hold for the pseudo block numerical range; here we only have the inclusion W n (L) ⊇ λ ∈ : 0 ∈ W n (L(λ)) =:W n ,0 (L), n ∈ N, see Proposition 4.4. Therein we also assess two other candidates W n ,i (L) = ε>0 W n ε,i (L), i = 1, 2, for the pseudo block numerical range for which W n ε,1 (L) is defined by the scalar condition det L(λ) ( f i ) <ε and W n ε,2 (L) by restricting to diagonal perturbations B ∈ L(H) with B < ε. In fact, we show that (1.5) and that, like the pseudo numerical range, the pseudo block numerical range W n (L) has the spectral inclusion property, i.e.
but, in general, none of the subsets of W n (L) in (1.5) is large enough to contain σ ap (T ), see Example 4.5.
Our second main result concerns the most important case n = 2, the so-called quadratic numerical range and pseudo quadratic numerical range. Here we prove a novel type of spectral inclusion for diagonally dominant and off-diagonally dominant L(λ) = (L i j (λ)) 2 i, j=1 in terms of the pseudo numerical ranges of the Schur complements S 1 , S 2 and, further, the pseudo quadratic numerical range of L, , and similarly for S 2 with the indices 1 and 2 reversed. For symmetric and anti-symmetric corners, i.e. L 21 (λ) ⊆ ±L 12 (λ) * , λ ∈ , we even show that As an interesting consequence, we are able to establish spectral separation and inclusion theorems for unbounded 2×2 operator matrices A = (A i j ) 2 i, j=1 with 'separated' diagonal entries; here 'separated' means that the numerical ranges of A 11 and A 22 lie in half-planes and/or sectors in the right and left half-plane C + and C − , respectively, separated by a vertical strip S:={z ∈ C : δ < Re z < α} with δ < 0 < α around iR. More precisely, without any bounds on the order of diagonal dominance or off-diagonal dominance we show that, if ϕ, ψ ∈ [0, π 2 ] are the semi-angles of A 11 and A 22 and τ := max{ϕ, ψ}, then and σ (A) ⊆ if ρ(A) ∩ (C\ ) = ∅, see Theorem 6.1. This result is a great step ahead compared to the earlier result [27,Thm. 5.2] where the dominance order had to be restricted to 0.
Moreover, even to ensure the condition ρ(A) ∩ (C\ ) =∅ for the enclosure of the entire spectrum σ (A) in Theorem 6.1, we do not have to restrict the dominance order as usual for perturbation arguments. Our new weak conditions involve only products of the columnwise relative bounds δ 1 in the first and δ 2 in the second column, see Proposition 6.5; in particular, either δ 1 = 0 or δ 2 = 0 guarantees ρ(A) ∩ (C\ ) =∅ in Theorem 6.1 and hence σ ap (A) ⊆ .
As an application of our results, we consider abstract quadratic operator polynomials T (λ), λ ∈ C, induced by forms t(λ) = t 0 +2λa +λ 2 with dom t(λ) = dom t 0 , λ ∈ C, as they arise e.g. from linearly damped wave equations (1.6) where the non-negative potential q and damping a may be singular and/or unbounded, cf. [11,[13][14][15] where also accretive damping was considered, and for which it is wellknown that the spectrum is symmetric with respect to R and confined to the closed left half-plane.
Here we use a finely tuned assumption on the 'unboundedness' of a with respect to t 0 , namely p-subordinacy for p ∈ [0, 1), comp. [20, § 5.1] or [29,Sect. 3] for the operator case. More precisely, if t 0 ≥ κ 0 ≥ 0, a ≥ α 0 ≥ 0 with dom t 0 ⊆ dom a and there exist p ∈[0, 1) and C p > 0 with to prove that the non-real spectrum of T satisfies the bounds is either empty or it is confined to one bounded interval, to one unbounded interval or to the disjoint union of a bounded and an unbounded interval , see Theorem 7.1 and Figure 2. Moreover, we describe both the thresholds for the transitions between these cases and the enclosures for σ (T ) ∩ R precisely in terms of p, C p , κ and κ 0 . As a concrete example, we consider the damped wave equation (1.6) with , c 1 , c 2 ≥ 0 and r ∈ [0, 1). For the special case q(x) = |x| 2 , a(x) = |x| k , x ∈ R d , with k ∈[0, 2), the new spectral enclosure in Theorem 7.1 yields The paper is organised as follows. In Sect. 2 we introduce the pseudo numerical range of operator functions and form functions and study the relation of W (T ) and W (T ) ∩ . In Sect. 3 we establish spectral inclusion results in terms of the pseudo numerical range. In Sect. 4 we define the block numerical range W n (L) and pseudo block numerical range W n (L) of unbounded n × n operator matrix functions L, investigate the differences to the special case n = 1 of the pseudo numerical range W 1 (L) = W (L) and prove corresponding spectral inclusion theorems. In Sect. 5 we establish new enclosures of the approximate point spectrum of 2 × 2 operator matrix functions by means of the pseudo numerical ranges of their Schur complements. In Sect. 6 we apply them to prove spectral bounds for diagonally dominant and off-diagonally dominant operator matrices with symmetric or anti-symmetric corners without restriction on the dominance order. Finally, in Sect. 7, we apply our results to linearly damped wave equations with possibly unbounded and/or singular damping and potential.
Throughout this paper, H and H i , i = 1, . . . , n, denote Hilbert spaces, L(H) denotes the space of bounded linear operators on H and ⊆ C is a domain.

The Pseudo Numerical Range of Operator Functions and Form Functions
In this section, we introduce the new notion of pseudo numerical range for operator functions {T (λ) : λ ∈ } and form functions {t(λ) : λ ∈ }, respectively, briefly denoted by T and t if no confusion about can arise. While the values T (λ) and t(λ) may be bounded/unbounded linear operators and sesquilinear forms in a Hilbert space H, the notion of pseudo numerical range is new also in the bounded case. The numerical range of T and t, respectively, are defined as comp. [20, § 26]. In the simplest case of a monic linear operator polynomial T (λ) = T 0 − λI H , λ ∈ C, this notion coincides with the numerical range W (T 0 ) of the linear operator T 0 , and analogously for forms; note that the latter is also denoted by (T 0 ), e.g. in [17,Sect. V.3.2].
The following new concept of pseudo numerical range employs the notion of εpseudo numerical range W ε (T ), ε > 0, introduced in [10, Def. 4.1]; the equivalent original definition therein, see (2.1) below, was designed to obtain computable enclosures for spectra of rational operator functions.

Definition 2.1
We introduce the pseudo numerical range of an operator function T and a form function t, respectively, as Clearly, for monic linear operator polynomials T (λ) = A−λI H , λ ∈ C, the pseudo numerical range is nothing but the closure of the classical numerical range W (A) of the linear operator A, and analogously for forms.
The pseudo numerical range of operator or form functions, is, like their numerical ranges, in general neither convex nor connected, and, even for families of bounded operators or forms, it may be unbounded.
(ii) In general, the pseudo numerical range need neither be open nor closed in equipped with the relative topology, see Examples 3.2 (i) and 2.9, respectively. (iii) Neither the closures nor the interiors with respect to the relative topology on of the pseudo numerical range and the numerical range need to coincide, see Example 3.2 (i) and (ii).
The following alternative characterisation of the pseudo numerical range will be frequently used in the sequel.

Proposition 2.3 For every
and, consequently, Proof We show the claim for W ε (T ); then the claim for W (T ) is obvious by Definition 2.1. The proof for W ε (t) and W (t) is analogous. Let ε > 0 be arbitrary and λ ∈ W ε (T ). There exists a bounded operator B in H with B < ε such that λ ∈ W (T + B), i.e.
The following properties of the pseudo numerical range with respect to closures, form representations and Friedrichs extensions are immediate consequences of its alternative description (2.2).
Here an operator A or a form a is called sectorial if its numerical range lies in a sector  (iii) The claim is a consequence of (i) and (ii).
The alternative characterisation (2.2) might suggest that there is a relation between the pseudo numerical range W (T ) and the closure W (T ) ∩ of the numerical range W (T ) in . However, in general, there is no inclusion either way between them, see e.g. Example 3.2 where W (T ) W (T ) ∩ and Example 2.9 where W (T ) ∩ W (T ). In fact, it was already noted in [31,Prop. 2.9], for continuous functions of bounded operators and for the more general case of block numerical ranges, that, for λ ∈ , the converse holds only under additional assumptions. More precisely, for families of bounded linear operators however, the following is known.

(i) If T is a (norm-)continuous family of bounded linear operators, then
(ii) If T is a holomorphic family of bounded linear operators and there exist k ∈ N 0 and μ ∈ with 3) The following simple example from [31,Ex. 2.11], which is easily adapted to the unbounded case, shows that condition (2.3) is essential for the equality W (T ) ∩ = W (T ) and for the spectral inclusion σ (T ) ⊆ W (T ) ∩ .
Further, it is easy to see that In the sequel we generalise Theorem 2.5 (i) and (ii) to families of unbounded operators and/or forms, including operator polynomials and sectorial families with constant form domain. In the remaining part of this section, we study the relation between W (T ) and W (T ) ∩ ; results containing spectral enclosures may be found in Sect. 3.

Proposition 2.7 Let T be an operator polynomial in
and analogously for form polynomials.
has degree n for each m ∈ N. Let λ m 1 , . . . , λ m n ∈ C denote its zeros. Then λ m j ∈ W (T ), j = 1, . . . , n, and p m admits the factorisation Next we generalise Theorem 2.5 (i) to families of sectorial forms with constant domain which satisfy a natural continuity assumption, see [17,Thm. VI.3.6]. This assumption is met, in particular, by holomorphic form families of type (a) and associated operator families of type (B).
Recall that a family t of densely defined closed sectorial sesquilinear forms in H is called holomorphic of type (a) if its domain is constant and the mapping λ → t(λ)[ f ] is holomorphic for every f ∈ D t :=dom t(λ). The associated family T of m-sectorial operators is called holomorphic of type (B), see [17,Sect. VII.4.2] and also [30]. Sufficient conditions on form families to be holomorphic of type (a) can be found in [17,§VII.4].

Theorem 2.8 Let t be a family of sectorial sesquilinear forms in
for all λ ∈ B r (λ 0 ) and f ∈ D t . Then

In particular, if t is a holomorphic form family of type (a) with associated holomorphic operator family T of type (B) in H, then
Since |Re t(λ 0 )[ f n ]| ≤ |t(λ 0 )[ f n ]| and w(λ n ) → 0, n → ∞, we obtain that, for n ∈ N sufficiently large, Now suppose that t and T are holomorphic families of type (a) and (B), respectively. We only need to show the second inclusion, the first one then follows from W (T ) ⊆ W (t) and Corollary 2.4 (ii). The second inclusion follows from what we already proved since for holomorphic form families of type (a), after a possible shift t+c where c > 0 is sufficiently large to ensure Re t(λ 0 ) ≥ 1, [17, Eqn. VII.(4.7)] shows that assumption (2.4) is satisfied. Theorem 2.5 (i) does not extend to analytic families of sectorial linear operators with non-constant form domains, as the following example inspired by [17, Ex. VII. 1.4] illustrates.

Spectral Enclosure via Pseudo Numerical Range
In this section we derive spectral enclosures for families of unbounded linear operators T (λ), λ ∈ , using the pseudo numerical range W (T ). The latter is tailored to enclose the approximate point spectrum.
The spectrum and resolvent set of an operator family T (λ), λ ∈ , respectively, are defined as and analogously for the various subsets of the spectrum. In addition to the approximate point spectrum we introduce the ε-approximate point spectrum, see [22] for the operator case, The latter is a subset of the ε-pseudo spectrum which was defined for operator functions with unbounded closed values in [8, Sect. 9.2, (9.9)], comp. also [7]. Clearly, for monic linear polynomials T (λ) = A − λI H , λ ∈ C, these notions coincide with the spectrum, resolvent set, approximate point spectrum, ε-approximate point spectrum and ε-pseudo spectrum of the linear operator A.

Proposition 3.1 For any operator family T (λ), λ ∈ , and every
and hence Proof The claims follow easily from (3.1) and Definition 2.1 together with Cauchy-Schwarz' inequality and (2.1) in Proposition 2.3.
The following simple examples illustrate some properties of the set W (T ) versus W (T ) ∩ , in particular, in view of spectral enclosures.
Then, for the nonholomorphic family T (λ) = A+|sin λ|, λ ∈ := C, it is easy to see that In fact, the claims for W (T ) are obvious.
Moreover, for arbitrary h ∈ H, h = 0, In the following, we generalise the spectral enclosure for bounded holomorphic families in Theorem 2.5 (ii) to holomorphic form families t of type (a) and associated operator families of type Here, for k ∈ N 0 , we denote the k-th derivative of t by

Theorem 3.3 Let t be a holomorphic form family of type (a) with associated holomorphic operator family T of type (B) in
If, in addition, the operator family T has constant domain, then

Remark 3.4 (i) Since t(λ)
is densely defined, closed and sectorial for all λ ∈ , condition (3.2) for k = 0 has the two equivalent forms hence, by Proposition 2.3 a sufficient condition for (3.2) is (ii) For operator polynomials T , which are holomorphic and have constant domain by definition, see Proposition 2.7, no sectoriality assumption is needed for the enclosure By Propositions 2.7 and 3.1, the above holds under the mere assumption that 0 / ∈ W (A n ) where A n is the leading coefficient of T ; note that then (3.2) holds with k = n and arbitrary μ ∈ C. This generalises the classical result [20,Thm. 26.7] To prove the claim stated at the beginning assume, to the contrary, that 0 ∈ W (t (k) (μ)), Since f n = 1, n ∈ N, it follows from (3.4) and the above inequality that there exists m n ≥ n such that In view of t (k) (μ)[ f n ] → 0, n → ∞, this implies the required claim This completes the proof that (3.2) holds with D t instead of D. By Corollary 2.4 (ii), we have W (t) = W (T ) ⊆ . Thus, due to (2.5), for the claimed equalities between pseudo numerical and numerical ranges it is sufficient to show W (t) ⊆ W (t) and W (t) ⊆ W (T ), respectively.
The enclosures of the spectrum follow from Proposition 3.1 and from the fact that σ (T (λ)) ⊆ W (T (λ)) since T (λ) is m-sectorial for all λ ∈ .
As forms are the natural objects regarding numerical ranges, it is not surprising that the inclusion W (T ) ⊆ W (T ) ∩ in Theorem 3.3 might cease to hold for more general analytic operator families where the connection to a family of forms is lost. Nevertheless, using an analogous idea as in the proof of Theorem 3.3, one can prove the corresponding inclusion for the approximate spectrum.
Recall that an operator family T in H is called holomorphic of type (A) if it consists of closed operators with constant domain and for each f ∈ D T := dom T (λ), the mapping λ → T (λ) f is holomorphic on . Here, for k ∈ N 0 , the k-th derivative of T is defined as

Pseudo Block Numerical Ranges of Operator Matrix Functions and Spectral Enclosures
In this section we introduce the pseudo block numerical range of n ×n operator matrix functions for which the entries may have unbounded operator values. While we study its basic properties for n ≥ 2, we study the most important case n = 2 in greater detail.
We suppose that with respect to a fixed decomposition H = H 1 ⊕ · · · ⊕ H n with n ∈ N, a family L = {L(λ) : λ ∈ } of densely defined linear operators in H admits a matrix representation here L i j are families of densely defined and closable linear operators from H j to H i , i, j = 1, . . . , n, and dom L(λ) = D 1 (λ) ⊕ · · · ⊕ D n (λ), The following definition generalises, and unites, several earlier concepts: the block numerical range of n × n operator matrix families whose entries have bounded linear operator values, see [23], the block numerical range of unbounded n × n operator matrices, see [24], and in the special case n = 2, the quadratic numerical range for bounded analytic operator matrix families and unbounded operator matrices, see [28] and [19], [27], respectively. Further, we introduce the new concept of pseudo block numerical range.
(ii) We introduce the pseudo block numerical range of L as Note that, indeed, if L(λ) = A−λI H , λ ∈ C, with an (unbounded) operator matrix A in H, then dom L(λ) = dom A is constant for λ ∈ C and W n (L) coincides with the block numerical range W n (A) first introduced in [24] and, for n = 2, in [27]. While the pseudo numerical range also satisfies W (L) = W (L) = W (A) this is no longer true for the pseudo block numerical range when n > 1; in fact, Example 4.5 below shows that W 2 (L) = W 2 (L) = W 2 (A) is possible.

Remark 4.2
It is not difficult to see that, for the block numerical range and the pseudo block numerical range of general operator matrix families, and W n (L) ⊆ W n (L). If dom L(λ) =: D L , λ ∈ , is constant, we can also write There are several other possible ways to define the pseudo block numerical range. In the following we show that, in general, they inevitably fail to contain the approximate point spectrum of an operator matrix family. While for the pseudo numerical range, analogous concepts as in Definition 4.3 coincide by Proposition 2.3, this is not true for the pseudo block numerical range. Here, in general, we only have the following inclusions.

Proposition 4.4 The pseudo block numerical range W n (L) satisfies
Proof We consider the case n = 2; the proofs for n > 2 are analogous. The leftmost and rightmost inclusions are trivial by definition. For the remaining inclusions, it is sufficient to show that, for every ε > 0, Then the respective claims follow by taking the intersection over all ε > 0. Let ε > 0 and λ ∈ W 2 ε,1 (L). Then there exists f ∈ dom L(λ) ∩ S 2 with

Now the first inclusion in (4.3) follows from
For the second inclusion, let λ ∈ with dist(0, W 2 (L(λ))) < √ ε, i.e. there exists μ ∈ C, |μ| < √ ε, with μ ∈ W 2 (L(λ)) or, equivalently, 0 ∈ W 2 (L(λ)−μI H ). By (4.1), the latter is in turn equivalent to Clearly, in the simplest case L(λ) = A − λI H , λ ∈ C, with an n × n operator matrix A in H we have this shows that W n ,0 (L) fails to enclose the spectrum of L whenever W n (A) does. The following example shows that, already in this simple case, in fact none of the subsets W n ,1 (L) ⊆ W n ,0 (L) ⊆ W n ,2 (L) of the pseudo block numerical range W n (L), see (4.2), is large enough to contain the approximate point spectrum σ ap (L). Clearly, W 2 (L) = W 2 (A) = {0}. We will now show that

Example 4.5 Let
By the definition of W 2 ,2 (L) and since W 2 ε,2 (L) ⊆ B ε (0), ε > 0, it follows that W 2 ,2 (L) = {0} which, together with (4.2), proves the first three equalities. To prove the two equalities on the right, and hence the claimed inequality, let λ ∈ C be arbitrary. If λ = 0, then λ ∈ W 2 (L) by (4.3). If λ = 0, we define the bounded operator matrices where δ mk denotes the Kronecker delta. Then B k → 0 as k → ∞ and a straightforward calculation shows that On the one hand, for arbitrary ε > 0, this implies that there exists N ∈ N such that and thus λ ∈ W 2 (L) by intersection over all ε > 0. On the other hand, λ ∈ σ ap (L) With one exception, we now focus on the most important case n = 2 for which the notation is more customary. We establish various inclusions between the (pseudo) quadratic numerical range W 2 ( ) (L) and the (pseudo) numerical ranges of the diagonal operator functions A, D, as well as between W 2 ( ) (L) and the (pseudo) numerical ranges of the Schur complements of L.
Proof The claims for the quadratic numerical range are consequences of (4.1) and of the corresponding statements [27, Prop. 3.2, 3.3 (i),(ii)] for operator matrices. So it remains to prove the claims (i) and (ii) for the pseudo quadratic numerical range; the proof of claim (iii) is completely analogous. (i) The inclusion for the quadratic numerical range in (i) applied to L + B with B < ε yields W 2 ε (L) ⊆ W ε (L) for any ε > 0. The claim for the pseudo quadratic numerical range follows if we take the intersection over all ε >0.
Both qualitative and quantitative behaviour of operator matrices are closely linked to the properties of their so-called Schur complements, see e.g. [27]; the same is true for operator matrix functions, see e.g. [28] for the case of bounded operator values.
of linear operators in H 1 and H 2 , respectively, with domains The following inclusions between the numerical ranges and pseudo numerical ranges of the Schur complements S 1 , S 2 and the quadratic numerical range and pseudo quadratic numerical range, respectively, of L hold.

Proposition 4.8 The numerical ranges and pseudo numerical ranges of the Schur complements satisfy
Proof The first claim follows from (4.1) and the corresponding statement [26,Thm. 2.5.8] for unbounded operator matrices. Using the first claim, the second claim can be proven in a similar way as the claim for the pseudo numerical range in Proposition 4.6 (ii).
The following spectral enclosure properties of the block numerical range and pseudo block numerical range hold for operator matrix functions. They generalise results for the case of bounded operator values from [31], see also [28] for n = 2, as well as the results for the operator function case, i.e. n = 1, in Proposition 3.1.

Proposition 4.9
Let L be a family of operator matrices. Then Proof The proof of the first inclusion is analogous to the bounded case, see [31,Thm. 2.14] and hence if, for all λ ∈ , σ (L(λ)) ⊆ W n (L(λ)), then Proof First let λ ∈ σ ap,ε (L). Then there exists f ε ∈ dom L(λ), f ε = 1, with L(λ) f ε <ε. The linear operator in H given by . By Proposition 4.9 and since B <ε, we conclude that λ ∈ W n (L − B) ⊆ W n ε (L), which proves the first claim.
The resolvent estimate in (4.5) follows from the first claim and from the definition of σ ap,ε (L), cf. the proof of Proposition 3.1.
Taking the intersection over all ε > 0 in the first claim, we obtain the inclusion σ ap (L) ⊆ W n (L).
Finally, the assumption that σ (L(λ)) ⊆ W n (L(λ)) for all λ ∈ implies that σ (L) ⊆ W n ,0 (L), see Definition 4.3. Now the second inequality in the last claim follows from the inclusion W n ,0 (L) ⊆ W n (L) by Proposition 4.4.

Spectral Enclosures by Pseudo Numerical Ranges of Schur Complements
In this section we establish a new enclosure of the approximate point spectrum of an operator matrix family L by means of the pseudo numerical ranges of the associated Schur complements and hence, by Proposition 4.8, in W 2 ,2 (L) and in the pseudo quadratic numerical range W 2 (L). Compared to earlier work, we no longer need restrictive dominance assumptions.
Theorem 5.1 Suppose that L is a family of 2 × 2 operator matrices as in (4.4). If λ ∈ σ ap (L)\(σ (A) ∪ σ (D)) is such that one of the conditions

Now assume that λ satisfies (ii). Since C(λ) is invertible, (5.3) shows that
Inserting (5.5) into (5.2) and using dom C(λ) ⊆ dom A(λ), we obtain that Since C(λ) −1 is bounded, we have C(λ) −1 k n → 0, n → ∞. Thus inf n∈N u n > 0 and (5.5) show that, without loss of generality, we can assume that inf n∈N w n > 0. Set By (ii) A(λ)C(λ) −1 is bounded and so A(λ)C(λ) −1 k n → 0, n → ∞. Now (5.6) yields S 1 (λ)w n → 0 and thus S 1 (λ)g n → 0, n → ∞, which proves λ ∈ σ ap (S 1 ). Finally, the first inclusion in (5.1) is obvious from what was already shown; the second inclusion in (5.1) follows from Proposition 3.1 and the last two inclusions from Proposition 4.8. For operator matrix families L with off-diagonal entries that are symmetric or antisymmetric to each other, we now establish conditions ensuring that the approximate point spectrum of L is contained in the union of the approximate point spectrum of one Schur complement and the pseudo numerical range of the corresponding diagonal entry, i.e. S 1 and D or S 2 and A. if dim H 1 > 1, then Note that here we do not assume that the entries of L are holomorphic. In the next section Theorem 5.3 will be applied with B(λ) = e iω(λ) B and C(λ) = e −iω(λ) C, where C ⊆ B * are constant and ω is real-valued, see the proof of Theorem 6.1.
The following corollary is immediate from Theorem 5.3 due to Proposition 4.6 and Proposition 4.8.

Proof of Theorem 5.3.
We only prove (i); the proof of (ii) is analogous. Let λ ∈ σ ap (L)\σ (D). In the same way as at the beginning of the proof of Theorem 5.1 we conclude that if lim inf n→∞ u n > 0, then λ ∈ σ ap (S 1 ). It remains to be shown that in the case lim inf n→∞ v n > 0, without loss of generality inf n∈N v n > 0, it follows that λ ∈ W (D).
Taking the scalar product with u n in (5.2) and with v n in (5.3), respectively, we conclude that (5.10) By subtracting from (5.9), or adding to (5.9), the complex conjugate of (5.10), we deduce that Taking real parts and using the accretivity of A(λ) and ∓D(λ), we obtain Since ∓D(λ) is sectorial with vertex 0 by assumption, this implies (∓D(λ)v n , v n ) → 0 and hence (D(λ)v n , v n ) → 0, n → ∞, which proves that λ ∈ W (D) by Proposition 2.3. Finally, the first inclusion in (5.7) is obvious from what was already proved; the second inclusion in (5.7) follows from Proposition 3.1. The last claim in (5.8) is then a consequence of Propositions 4.6 (iii) and 4.8.

Application to Structured Operator Matrices
In this section, we apply the results of the previous section to prove new spectral enclosures and resolvent estimates for non-selfadjoint operator matrix functions exhibiting a certain dichotomy. More precisely, we consider a linear monic family L(λ) = A − λI H , λ ∈ C, with a densely defined operator matrix We assume that the entries of A are densely defined closable linear operators acting between the respective spaces H 1 and/or H 2 , and that A, −D are accretive or even sectorial with vertex 0. This means that their numerical ranges lie in closed sectors ω with semi-axis R + and semi-angle ω = π/2 or ω ∈ [0, π/2), respectively, given by here arg : C → (−π, π] is the argument of a complex number with arg 0 = 0. The next theorem no longer requires bounds on the dominance orders among the entries in the columns of A, in contrast to earlier results in [27,Thm. 5.2] where the relative bounds had to be 0. The proof of Theorem 6.1 relies on Theorems 5.1 and 5.3, and on the following enclosures for the pseudo numerical ranges of the Schur complements. Lemma 6.2 Let A be as in (6.1) with C ⊆ B * and let λ ∈ C.
We show that assumptions (i) or (iii) imply (6.3); the proof when assumptions (ii) or (iv) hold is analogous.

Application to Damped Wave Equations in R d with Unbounded Damping
In this section we use the results obtained in Sect. 3 to derive new spectral enclosures for linearly damped wave equations with non-negative possibly singular and/or unbounded damping a and potential q. Our result covers a new class of unbounded dampings which are p-subordinate to − + q, a notion going back to [18, §.7  where t 0 and a are densely defined sesquilinear forms in H such that t 0 is closed, t 0 ≥ κ 0 ≥ 0, a ≥ α 0 ≥ 0 and dom t 0 ⊆ dom a. Suppose that there exist κ ≤ κ 0 and p ∈ (0, 1) such that a is p-form-subordinate with respect to t 0 − κ ≥ 0, i.e. there is in particular, D ≤0 = ∅ implies W (t) ∩ R = ∅. An elementary analysis shows that d is either identically zero, has no zero, one simple zero or two (possibly coinciding) zeros on [0, ∞), which we denote by x + and x − ≤ x + , respectively, if they exist. Then Which case prevails for fixed p ∈[0, 1) can be characterised by means of inequalities involving the constants κ 0 , κ and C p . For estimating λ 0 in (7.4) while respecting the restrictions in (7.5), we consider the functions It is easy to check that f + is monotonically increasing in s and monotonically decreasing in t, while f − is monotonically decreasing in s and monotonically increasing in t and hence, since s ≤ C p (t − κ) p , f + (s, t) ≤ f + (C p (t − κ) p , t)=:g + (t), f − (s, t) ≥ f − (C p (t − κ) p , t)=:g − (t). (7.8) Now we distinguish the two qualitatively different cases (7.6) and (7.7). To obtain the claimed enclosures for W (t) ∩ R, we use (7.5), (7.4) and (7.8) to conclude that g − (t) ≤ λ 0 ≤ g + (t) for some t ∈ D ≤0 . If (7.6) holds, there are the following two possibilities: (1) If d has no zeros on [0, ∞) or if d has at least one zero and x + <κ 0 , then D ≤0 = ∅ and thus W (t) ∩ R = ∅.
(2) If d has at least one zero x + and x + ≥ κ 0 , then D ≤0 is one bounded interval and W (t)∩ R ⊆ s − , s + , s − := min where in the latter case t 0 = max k(2 − k)