BIRKHOFF-JAMES ǫ-orthogonality SETS IN NORMED LINEAR SPACES

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1 BIRKHOFF-JAMES ǫ-orthogonality SETS IN NORMED LINEAR SPACES MILTIADIS KARAMANLIS AND PANAYIOTIS J. PSARRAKOS Dedicated to Profess Natalia Bebiano on the occasion of her 60th birthday Abstract. Consider a complex nmed linear space (X, ), and let χ, ψ X with ψ 0. Motivated by a recent wk of Chianopoulos and Psarrakos (011) on rectangular matrices, we introduce the Birkhoff- James ǫ-thogonality set of χ with respect to ψ, and exple its rich structure. 1. Introduction The numerical range (also known as the field of values) of a square complex matrix A n n is defined as F(A) x Ax : x n, x x 1} [8]. This range is a non-empty, compact and convex subset of, which has been studied extensively and is useful in understanding matrices and operats; see [, 3, 8, 10] and the references therein. The numerical range F(A) is also written in the fm (see [3, 10]) F(A) µ : A λi n µ λ, λ }, where denotes the spectral matrix nm (i.e., that nm subdinate to the euclidean vect nm) and I n is the n n identity matrix. As a consequence, F(A) is an infinite intersection of closed (circular) disks D (λ, A λi n ) µ : µ λ A λi n } (λ ), namely, (1.1) F(A) λ C µ : µ λ A λi n } λ C D (λ, A λi n ). F two elements χ and ψ of a complex nmed linear space (X, ), χ is said to be Birkhoff-James thogonal to ψ, denoted by χ BJ ψ, if χ+λψ χ f all λ [1, 9]. This thogonality is homogeneous, but it is neither symmetric n additive [9]. Meover, f any ǫ [0,1), χ is called Birkhoff- James ǫ-thogonal to ψ, denoted by χ ǫ BJ ψ, if χ + λψ 1 ǫ χ f 000 Mathematics Subject Classification. 46B99, 47A1. Key wds and phrases. nm, Birkhoff-James thogonality, Birkhoff-James ǫ- thogonality, numerical range, inner product. 1

2 MILTIADIS KARAMANLIS AND PANAYIOTIS J. PSARRAKOS all λ [4, 7]. It is wth mentioning that this relation is also homogeneous. In an inner product space (X,, ), with the standard thogonality relation, a χ X is called ǫ-thogonal to a ψ X, denoted by χ ǫ ψ, if χ,ψ ǫ χ ψ. Furtherme, χ ψ (resp., χ ǫ ψ) if and only if χ BJ ψ (resp., χ ǫ BJ ψ) [4, 7]. Inspired by (1.1) and the above definition of Birkhoff-James ǫ-thogonality, Chianopoulos and Psarrakos [6] (see also [5] f a primer wk) proposed the following definition f rectangular matrices: F any A,B n m with B 0, any matrix nm, and any ǫ [0, 1), the Birkhoff-James ǫ-thogonality set of A with respect to B is defined as (1.) F ǫ (A;B) µ : B ǫ BJ (A µb)} µ : A λb 1 ǫ B µ λ, λ } λ C D ( ) A λb λ,. 1 ǫ B The Birkhoff-James ǫ-thogonality set is a direct generalization of the standard numerical range. In particular, f n m,, B I n and ǫ 0, we have F 0 (A;I n ) F(A); see (1.1) and (1.). Meover, F ǫ (A;B) is a non-empty, compact and convex subset of that lies in the closed disk D ( 0, A 1 ǫ B ) and has interesting geometric properties [6]. In this note, we adopt ideas and techniques from [6] to introduce and study the Birkhoff-James ǫ-thogonality set of elements of a complex nmed linear space, generalizing results of [6]. In the next section, we give the definition of the set, and verify that it is always non-empty. In Section 3, we exple the growth of the set, and in Section 4, we derive characterizations of its interi and boundary. Finally, in Section 5, we describe the Birkhoff-James ǫ-thogonality set when the nm is induced by an inner product.. The definition Consider a complex nmed linear space (X, ) (f simplicity, X), and let χ,ψ X with ψ 0. F any ǫ [0,1), the Birkhoff-James ǫ-thogonality set of χ with respect to ψ is defined and denoted by (.1) F ǫ (χ;ψ) µ : ψ ǫ BJ (χ µψ)}.

3 BIRKHOFF-JAMES ǫ-orthogonality SETS 3 It is straightfward to see that F ǫ (χ;ψ) (.) (.3) µ : ψ λ(χ µψ) 1 ǫ ψ, λ } µ : ψ 1 λ (χ µψ) } 1 ǫ ψ, λ \ 0} 1 µ : λ λψ (χ µψ) } 1 ǫ ψ, λ \ 0} µ : χ (µ λ)ψ 1 ǫ ψ λ, λ } µ : χ λψ 1 ǫ ψ µ λ, λ } λ C D ( ) χ λψ λ,. 1 ǫ ψ The defining fmula (.3) implies that F ǫ (χ;ψ) is a compact and convex ( ) subset of, which lies in the closed disk D 0,. Furtherme, it is χ 1 ǫ ψ apparent that f any 0 ǫ 1 < ǫ < 1, F ǫ1 ǫ (χ;ψ) F By Collary. of [9], it follows that F ǫ (χ;ψ). (χ;ψ) is always non-empty. F clarity, we give a sht proof, adopting arguments from the proofs of Theem.1, Theem. and Collary. of [9]. Proposition.1. F any χ,ψ X with ψ 0, and any ǫ [0,1), the Birkhoff-James ǫ-thogonality set F ǫ (χ;ψ) is non-empty. Proof. Since F 0 (χ;ψ) F ǫ (χ;ψ) f every ǫ [0,1), it is enough to prove that F 0 (χ;ψ). Applying the Hahn-Banach Theem one can verify that f any nonzero ψ X, there is a linear functional T : X such that T(ψ) T ψ. As a consequence, and hence, T ψ T(ψ) T(ˆχ + ψ) T ˆχ + ψ, (.4) ψ BJ ˆχ, ˆχ Ker(T). ˆχ Ker(T), F the scalar µ T(χ) T ψ, we have that T(χ µψ) 0, and thus, χ µψ Ker(T). By (.4), ψ BJ (χ µψ), and hence, µ F 0 (χ;ψ). Next we derive some basic properties of the Birkhoff-James ǫ-thogonality set. Proposition.. Let χ,ψ X with ψ 0, and let ǫ [0,1). Then, f any nonzero b, F ǫ (χ;bψ) 1 b F ǫ (χ;ψ).

4 4 MILTIADIS KARAMANLIS AND PANAYIOTIS J. PSARRAKOS Proof. By the defining fmula (.1) of the Birkhoff-James ǫ-thogonality set F ǫ (χ;ψ) and the homogeneity of the Birkhoff-James ǫ-thogonality, it is straightfward that F ǫ (χ;bψ) µ : ψ ǫ BJ (χ (bµ)ψ)}. Proposition.3. Let χ and ψ be two nonzero elements of X. Then, f any ǫ [0,1), } µ 1 : µ F ǫ χ (χ;ψ), µ F ǫ ψ (ψ;χ). Proof. Consider a µ F ǫ χ (χ;ψ) with µ ψ. Then, by (.), we have λ ψ 1 λ χ 1 ǫ ψ λ µ λ 1, λ \ 0}, ψ λχ 1 ǫ ψ µ µ 1 λ 1 ǫ χ µ 1 λ, λ. Thus, µ 1 lies in F ǫ (ψ;χ). Proposition.4. Let a and b be two equivalent nms acting in X, and suppose that f two real numbers C,c > 0, c ζ a ζ b C ζ a f all ζ X. Then, f any χ,ψ X with ψ 0 and any ǫ [0,1), it holds that where ǫ F ǫ a (χ;ψ) F ǫ b (χ;ψ), 1 c (1 ǫ ) C. Proof. Suppose µ F ǫ a (χ;ψ). Then, it follows readily that χ λψ b χ λψ a 1 ǫ ψ a µ λ, λ, χ λψ b 1 ǫ c C ψ b µ λ, λ, and the proof is complete. 1 1 c (1 ǫ ) C F example, we consider the vects χ ψ b µ λ, λ, 1 + i 11i, ψ 1 + i + i i 3, and recall that the (equivalent in 3 ) nms and 1 satisfy ζ ζ 1 3 ζ f all ζ 3. The Birkhoff-James ǫ-thogonality sets F (χ;ψ) and F 0.75 F (χ;ψ), (χ;ψ) are estimated by the unshaded regions in the left,

5 BIRKHOFF-JAMES ǫ-orthogonality SETS 5 middle and right parts of Figure 1, respectively. Each estimation results from having drawn 000 circles of the fm µ : µ λ χ λψ }; see (.) and (.3). The compactness and the convexity of the sets are apparent, and since , Proposition.4 is also confirmed Imaginary Axis 0 Imaginary Axis 0 Imaginary Axis Real Axis Real Axis Real Axis Figure 1. The sets F 0.5 (χ;ψ) (left), F (χ;ψ) (middle), and F 0.75 (χ;ψ) (right) On the growth of F ǫ (χ;ψ) As mentioned befe, f 0 ǫ 1 < ǫ < 1 and f any two elements χ and ψ of a complex nmed linear space X with ψ 0, it holds that F ǫ1 (χ;ψ) F ǫ (χ;ψ). Theem 3.1. (F matrices, see [6, Proposition ].) Let χ,ψ X with ψ 0, and suppose that χ is not a scalar multiple of ψ. Then, f any 0 ǫ 1 < ǫ < 1, F ǫ1 ǫ (χ;ψ) lies in the interi of F (χ;ψ). Proof. It is enough to prove that f any µ F ǫ1 (χ;ψ), there is a real ρ µ > 0 such that the disk D(µ,ρ µ ) lies in F ǫ (χ;ψ). By the defining fmula (.) of the Birkhoff-James ǫ-thogonality set F ǫ ǫ1 (χ;ψ), f any µ F (χ;ψ), χ λψ 1 ǫ 1 ψ µ λ, λ, equivalently, χ µψ + (µ λ)ψ 1 ǫ 1 ψ µ λ, λ.

6 6 MILTIADIS KARAMANLIS AND PANAYIOTIS J. PSARRAKOS As a consequence, χ µψ + λψ 1 ǫ 1 ψ λ > 1 ǫ ψ λ, λ. Thus, f every complex number λ 0, ) χ µψ + λψ 1 ǫ (1 ψ λ ǫ 1 1 ǫ ψ λ > 0. Since χ is not a scalar multiple of ψ, it follows that χ µψ + λψ > 0, and hence, the continuous function f(λ) χ µψ + λψ 1 ǫ ψ λ takes only positive values in the disk D(0, 1). Thus, inf f(λ) min f(λ) > 0. λ D(0,1) λ D(0,1) This means that we can consider a real δ > 0 such that ) } δ min min (1 f(λ), ǫ 1 1 ǫ ψ. λ D(0,1) F every λ with λ > 1, we have ) χ µψ + λψ 1 ǫ (1 ψ λ ǫ 1 1 ǫ ψ δ, and thus, δ inf λ C As a consequence, f every ξ D χ µψ + λψ ( ) δ 0, ψ, 1 ǫ ψ λ }. χ (µ + ξ)ψ + λψ χ µψ + λψ ξψ ( ) δ Hence, D 0, ψ F ǫ (χ;ψ), and the proof is complete. 1 ǫ ψ λ, λ. Collary 3.. Let χ,ψ X with ψ 0, and suppose that χ is not a scalar multiple of ψ. Then, f any ǫ (0,1), F ǫ (χ;ψ) has a non-empty interi. Proposition 3.3. Let χ,ψ X with ψ 0. Then, χ aψ f some a if and only if F ǫ (χ;ψ) a} f every ǫ [0,1). Proof. If χ aψ f some a, then (.) yields F ǫ (χ;ψ) F ǫ (aψ;ψ) µ : (a λ)ψ 1 ǫ ψ µ λ, λ } µ : a λ 1 ǫ µ λ, λ }. It is apparent that a F ǫ (aχ;ψ). Furtherme, f any µ a and λ a, we have 0 a λ < 1 ǫ µ λ, i.e., µ / F ǫ (aψ;ψ).

7 BIRKHOFF-JAMES ǫ-orthogonality SETS 7 F the converse, suppose that F ǫ (χ;ψ) a} f an ǫ (0,1). Then, by Collary 3., χ is a scalar multiple of ψ, i.e., there is a b such that χ bψ. As a consequence, a λ 1 ǫ b λ, λ. F λ a, it follows that b a 0, and the proof is complete. By the proof of the previous proposition, it is clear that if F ǫ (χ;ψ) a} f an ǫ (0,1), then χ aψ, and consequently, F ǫ (χ;ψ) a} f all ǫ [0,1). Proposition 3.4. Let χ,ψ X with ψ 0, and let ǫ [0,1). Then, f any a,b, F ǫ (aχ + bψ;ψ) af ǫ (χ;ψ) + b. Proof. If a 0, then Proposition 3.3 yields F ǫ (aχ;ψ) 0} 0F ǫ (ψ;ψ). If a 0, then F ǫ (aχ;ψ) µ : aχ λψ 1 ǫ ψ µ λ, λ } µ : χ λ a ψ } 1 ǫ ψ µ, λ µ : χ λψ 1 ǫ ψ af ǫ (χ;ψ). Furtherme, f any a,b, F ǫ (aχ + bψ;ψ) a λ a µ a λ, λ } µ : aχ + (b λ)ψ 1 ǫ ψ µ λ, λ } µ : aχ λψ 1 ǫ ψ (µ b) λ), λ } µ : µ b F ǫ (aχ;ψ) }, and the proof is complete. If we allow the value ǫ 1, then (.) implies that F 1 (A;B). Furtherme, if χ is not a scalar multiple of ψ, then F ǫ (A;B) can be arbitrarily large f ǫ sufficiently close to 1. Theem 3.5. (F matrices, see [6, Proposition 4].) Let χ,ψ X with ψ 0, and suppose that χ is not a scalar multiple of ψ. Then, f any bounded region Ω, there is an ǫ Ω [0,1) such that Ω F ǫω (χ;ψ). Proof. Without loss of generality, we may assume that the region Ω is compact. F the sake of contradiction, we also assume that f every ǫ [0,1), there is scalar µ ǫ such that µ ǫ / F ǫ (χ;ψ). Then, there exist two sequences

8 8 MILTIADIS KARAMANLIS AND PANAYIOTIS J. PSARRAKOS ǫ n } n N [0,1) and µ n } n N Ω such that ǫ n 1 and µ n / F ǫn (χ;ψ) f all n Æ. By the compactness of Ω, it follows that µ n } n N has a converging subsequence, say µ kn } n N Ω, which converges to a µ Ω. If µ F ˆǫ (χ;ψ) f some ˆǫ [0,1), then by Theem 3.1, and without loss of generality, we may assume that µ lies in the interi of F ˆǫ (χ;ψ). Then there is an n Æ such that µ kn F ˆǫ (χ;ψ) f every n n. Meover, there is an n Æ such that ǫ kn > ˆǫ f every n n. As a consequence, f every n maxn,n }, µ kn F ˆǫ (χ;ψ) F ǫ kn (χ;ψ); this a a contradiction. So, f every ǫ [0,1), µ / F ǫ (χ;ψ). Thus, f every ǫ n 1 1 n, n Æ, there is a scalar λ n such that ( ) χ (µ λ n )ψ < n ψ λ n 1 n ψ λ n, (3.1) λ n ψ χ µψ λ n ψ χ µψ < 1 n ψ λ n, ( λ n ψ 1 1 ) < χ µψ. n Hence, f every n, λ n < χ µψ ψ ( ) 1 1 n χ µψ ψ. The bounded sequence λ,λ 3,... has a converging subsequence λ kn } n N which converges to a scalar λ 0. By (3.1), it follows and as n +, λ kn ψ χ µψ < 1 k n ψ λ kn, λ 0 ψ χ µψ 0. This is a contradiction because χ is not a scalar multiple of ψ. Collary 3.6. Let χ,ψ X with ψ 0. If χ is not a scalar multiple of ψ, then n NF 1 1 n (χ;ψ).

9 BIRKHOFF-JAMES ǫ-orthogonality SETS 9 4. The interi and the boundary of F ǫ (χ;ψ) Consider the Birkhoff-James ǫ-thogonality set F ǫ (χ;ψ), and denote its ] interi by Int [F ǫ (χ;ψ), and its boundary by F ǫ (χ;ψ). Proposition 4.1. Let χ,ψ X, with ψ 0. Then, f any ǫ [0,1), ] Int [F ǫ (χ;ψ) µ : χ λψ > 1 ǫ ψ µ λ, λ }. ] Proof. If µ Int [F ǫ (χ;ψ), then there is a real ρ > 0 such that µ + ρe iθ F ǫ (χ;ψ) f every θ [0,π]. Hence, f every λ, χ λψ 1 ǫ ψ µ + ρe iθ λ, Setting θ λ arg(µ λ), we observe that θ [0,π]. χ λψ 1 ǫ ψ µ + ρe iθ λ λ > 1 ǫ ψ µ λ, completing the proof. Theem 4.. (F matrices, see [6, Proposition 16].) Let χ, ψ X with ψ 0, and let ǫ [0,1). Suppose also that µ 0 F ǫ (χ;ψ). (i): The scalar µ 0 lies on the boundary F ǫ (χ;ψ) if and only if χ λψ } 1 ǫ ψ µ 0 λ 0. inf λ C (ii): If ǫ > 0, then µ 0 F ǫ (χ;ψ) if and only if χ λψ } 1 ǫ ψ µ 0 λ 0, min λ C equivalently, if and only if χ λ 0 ψ 1 ǫ ψ µ 0 λ 0 f some λ 0. Proof. (i) Suppose that µ 0 is a boundary point of the Birkhoff-James ǫ- thogonality set (recall (.3)) F ǫ (χ;ψ) ( ) χ λψ D λ,. λ C 1 ǫ ψ Then, f any δ > 0, there is a λ δ such that χ λ δ ψ < 1 ǫ ψ µ 0 λ δ + δ. Since the quantity χ λ δ ψ 1 ǫ ψ µ 0 λ δ is nonnegative, as δ 0 +, it follows that inf χ λψ 1 ǫ ψ µ 0 λ } 0. λ C

10 10 MILTIADIS KARAMANLIS AND PANAYIOTIS J. PSARRAKOS F the converse, we assume that inf χ λψ 1 ǫ ψ µ 0 λ } ] λ C 0 and µ 0 Int [F ǫ (χ;ψ). Then, by (.3), there exists a real ρ > 0 such that [ ( )] χ λψ D(µ 0,ρ) Int D λ,, λ. 1 ǫ ψ As a consequence, χ λψ 1 ǫ ψ µ 0 λ > 1 ǫ ψ ρ > 0, λ. This means that inf λ C χ λψ } 1 ǫ ψ µ 0 λ > 0 which is a contradiction. (ii) F every δ n 1 n (n Æ), there is a λ n such that χ λ n ψ < 1 ǫ ψ µ 0 λ n + δ n, χ λ n ψ < 1 ǫ ψ µ 0 λ n + 1 n, λ n ψ χ < 1 ǫ ψ ( µ 0 + λ n ) + 1 n. Since ǫ > 0, one can verify that λ n < χ + 1 ǫ ψ µ ψ ( 1 1 ǫ ), i.e., the sequence λ n } n N is bounded and has a converging subsequence λ kn λ 0. As a consequence, and as n +, χ λ kn ψ < 1 ǫ ψ µ 0 λ kn + 1 k n, n Æ, χ λ 0 ψ 1 ǫ ψ µ 0 λ 0. This inequality can hold only as an equality because µ F ǫ (χ;ψ), and the proof is complete. Proposition 4.1 and Theem 4. yield readily the following. Collary 4.3. Let χ,ψ X, with ψ 0. Then, f any ǫ (0,1), ] Int [F ǫ (χ;ψ) µ : χ λψ > 1 ǫ ψ µ λ, λ }.

11 BIRKHOFF-JAMES ǫ-orthogonality SETS The case of nms induced by inner products In the special case of nms induced by inner products, we can fully describe the Birkhoff-James ǫ-thogonality set F ǫ (χ;ψ). In particular, F ǫ (χ;ψ) is always a closed disk; this is the case f F 0.5 (χ;ψ) in the left part of Figure 1. Theem 5.1. (F matrices, see [6, Section 5].) Let χ,ψ X with ψ 0 and ǫ [0,1), and suppose that the nm is induced by an inner product,. Then the Birkhoff-James ǫ-thogonality set of χ with respect to ψ is the closed disk F ǫ (χ;ψ) D ( χ,ψ ψ, χ χ,ψ ψ ψ ) ǫ. 1 ǫ ψ Proof. A scalar µ lies in F ǫ (χ;ψ) if and only if [4, 7] equivalently, if and only if equivalently, if and only if ψ ǫ (χ µψ), ψ,χ µψ ǫ ψ χ µψ, ψ,χ µψ χ µψ,ψ ǫ ψ χ µψ,χ µψ, equivalently, if and only if χ,ψ ψ 4 µ ψ,χ χ,ψ µ ψ ψ + µ ǫ ( χ ψ µ ψ,χ ψ µ χ,ψ ) ψ + µ, equivalently, if and only if µ χ,ψ ψ (1 ǫ ) ǫ ψ χ χ,ψ ψ ψ. The proof is complete. References [1] G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J., 1 (1935), [] F.F. Bonsall and J. Duncan, Numerical Ranges of Operats on Nmed Spaces and of Elements of Nmed Algebras, London Mathematical Society Lecture Note Series, Cambridge University Press, New Yk, [3] F.F. Bonsall and J. Duncan, Numerical Ranges II, London Mathematical Society Lecture Notes Series, Cambridge University Press, New Yk, [4] J. Chmielinski, On an ε-birkhoff thogonality, J. Ineq. Pure and Appl. Math., 6 (005), Article 79. [5] Ch. Chianopoulos, S. Karanasios and P. Psarrakos, A definition of numerical range of rectangular matrices, Linear Multilinear Algebra, 57 (009),

12 1 MILTIADIS KARAMANLIS AND PANAYIOTIS J. PSARRAKOS [6] Ch. Chianopoulos and P. Psarrakos, Birkhoff-James approximate thogonality sets and numerical ranges, Linear Algebra Appl., 434 (011), [7] S.S. Dragomir, On approximation of continuous linear functionals in nmed linear spaces, An. Univ. Timişoara Ser. Ştiinţ. Mat., 9 (1991), [8] R.A. Hn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, [9] R.C. James, Orthogonality and linear functionals in nmed linear spaces, Trans. Amer. Math. Soc., 61 (1947), [10] J.G. Stampfli and J.P. Williams, Growth conditions and the numerical range in a Banach algebra, Tôhoku Math. Journ., 0 (1968), Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens, Greece address: kararemilt@gmail.com, ppsarr@math.ntua.gr