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1 Erratum to: Positivity (2013) 17:1–15 DOI 10.1007/s11117-011-0138-4
The original publication of this paper contained an error in Lemma 2(g). The statement (which is used later in Theorem 6, Lemma 12, and Theorem 11) is false in general.
We will show that \(\mathbf{D }_\mathfrak{w _{k}}\mathfrak t \uparrow \mathbf{D }_\mathfrak{w }\mathfrak{t }\) whenever \(\mathfrak w _k\uparrow \mathfrak w \) and \(\mathfrak w \) is dominated by \(\mathfrak t \). Since this condition is satisfied in Theorem 6, Lemma 12, and Theorem 11, the results remain true (with the same proof).
The first part of the following theorem is the reformulation of (Proposition 5.1 in [1]) for nonnegative sesquilinear forms, the second part is an easy consequence. This is the correct version of Lemma 2(g).
Theorem 1
Let \(\mathfrak t \), \(\mathfrak w \), and \(\mathfrak w _k~(k\in \mathbb N )\) be nonnegative sesquilinear forms on the complex linear space \(\mathfrak D \). Assume further that \(\mathfrak w _k\uparrow \mathfrak w \), and \(\mathfrak w \le \alpha \mathfrak t \) for some \(\alpha >0\). Then
Proof
Since \(\mathfrak w _k:\mathfrak t \le \mathfrak w :\mathfrak t \) for all \(k\in \mathbb N \), the supremum (denoted by \(\mathfrak s \))
exists. We will show that \(\mathfrak s =\mathfrak w :\mathfrak t \). Since
one can apply (Theorem 3.4 in [3]). I.e., the parallel difference \(\mathfrak s \div \mathfrak t \) exists and
Hence, according to (Proposition 2.7 in [2]) one obtain that
Since \(\mathfrak w _k\) is almost dominated by \(\mathfrak t \), it follows also that
Taking the supremum in \(k\), \(\mathfrak s \div \mathfrak t \ge \mathfrak w \) and
Consequently,
It remains only to show that \(\sup \nolimits _{k\in \mathbb N }\mathbf D _\mathfrak{w _k}\mathfrak t =\mathbf{D }_\mathfrak{w }\mathfrak{t }\). Observe first that the sequence \((n\mathfrak w _k)_{k\in \mathbb N }\) is also monotonically nondecreasing, \(\sup \nolimits _{k\in \mathbb N }n\mathfrak w _k=n\mathfrak w \), and \(n\mathfrak w \le n\alpha \mathfrak t \). Consequently, we can take the supremum in \(k\):
This gives immediately that \(\sup _{k\in \mathbb N }\mathbf D _\mathfrak{w _k}\mathfrak t \ge \mathbf{D }_\mathfrak{w }\mathfrak{t }\). Since the inequality \(\mathfrak w _k\le \mathfrak w \) implies
for all \(k\in \mathbb N \), the proof is complete. \(\square \)
Corollary 1
Let \(\mathfrak t \) and \(\mathfrak w \) be nonnegative sesquilinear forms on the complex linear space \(\mathfrak D \). Then
i.e., \(\mathbf{D }_\mathfrak{w }\mathfrak{t }\) is a \(\mathfrak t \)-quasi-unit for all \(\mathfrak w \).
References
Eriksson, S.L., Leutwiler, H.: A potential theoretic approach to parallel addition. Math. Ann. 274, 301–317 (1986)
Hassi, S., Sebestyén, Z., de Snoo, H.: Lebesgue type decompositions for nonnegative forms. J. Funct. Anal. 257(12), 3858–3894 (2009)
Sebestyén, Z., Titkos, T.: Parallel subtraction of nonnegative forms. Acta Math. Hung. 136(4), 252–269 (2012)
Acknowledgments
The authors wish to express their gratitude to Zsigmond Tarcsay for pointing out the error.
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The online version of the original article can be found under doi:10.1007/s11117-011-0138-4.
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Sebestyén, Z., Titkos, T. Erratum to: Complement of forms. Positivity 17, 941–943 (2013). https://doi.org/10.1007/s11117-013-0250-8
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DOI: https://doi.org/10.1007/s11117-013-0250-8