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We show that for every $0\\leq A_0\\in L^{r}_{n}(Y, Z)$, $|M_{A_0, B}|(T)=M_{A_0, |B|}(T)$ holds for all $B\\in L^{r}(W, X)$ and all $T\\in L^{r}_{n}(X, Y)$. Furthermore, if $W$, $X$, $Y$ and $Z$ are Dedekind complete Banach lattices such that $X$ and $Y$ have order continuous norms, then $|M_{A,\\, B}|=M_{|A|, \\,|B|}$ for all $ A\\in L^{r}(Y, Z)$ and all $B\\in L^{r}(W, X"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1609.06913","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-09-22T11:14:17Z","cross_cats_sorted":[],"title_canon_sha256":"b9db83b4d5876831672585806ccf911a7cbffc23788db9ab10a54b7fc0593cff","abstract_canon_sha256":"0935354e11ac9262c8d7f518eac25db788debaa1306cce40506c246718ba664b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:04.827719Z","signature_b64":"iHNPeLSgVMWW/+s3uxH973iXaESUTWQJK54BJmcbbE6Zob2JNMoP9Q5QHrUt6W8inS0RMfWBqL5+USFc9uEsDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4f65fc3ff7348150ef8baf03b49599a6cbc47a7f772d0dfc7dec69661db17a0d","last_reissued_at":"2026-05-18T01:04:04.827038Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:04.827038Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Two-sided multiplication operators on the space of regular operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Anton R. Schep, Jin Xi Chen","submitted_at":"2016-09-22T11:14:17Z","abstract_excerpt":"Let $W$, $X$, $Y$ and $Z$ be Dedekind complete Riesz spaces. For $A\\in L^{r}(Y, Z)$ and $B\\in L^{r}(W, X)$ let $M_{A,\\,B}$ be the two-sided multiplication operator from $L^{r}(X, Y)$ into $L^r(W,\\,Z)$ defined by $M_{A,\\,B}(T)=ATB$. We show that for every $0\\leq A_0\\in L^{r}_{n}(Y, Z)$, $|M_{A_0, B}|(T)=M_{A_0, |B|}(T)$ holds for all $B\\in L^{r}(W, X)$ and all $T\\in L^{r}_{n}(X, Y)$. Furthermore, if $W$, $X$, $Y$ and $Z$ are Dedekind complete Banach lattices such that $X$ and $Y$ have order continuous norms, then $|M_{A,\\, B}|=M_{|A|, \\,|B|}$ for all $ A\\in L^{r}(Y, Z)$ and all $B\\in L^{r}(W, X"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.06913","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1609.06913","created_at":"2026-05-18T01:04:04.827156+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.06913v1","created_at":"2026-05-18T01:04:04.827156+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.06913","created_at":"2026-05-18T01:04:04.827156+00:00"},{"alias_kind":"pith_short_12","alias_value":"J5S7YP7XGSAV","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_16","alias_value":"J5S7YP7XGSAVB34L","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_8","alias_value":"J5S7YP7X","created_at":"2026-05-18T12:30:22.444734+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/J5S7YP7XGSAVB34LV4B3JFMZU3","json":"https://pith.science/pith/J5S7YP7XGSAVB34LV4B3JFMZU3.json","graph_json":"https://pith.science/api/pith-number/J5S7YP7XGSAVB34LV4B3JFMZU3/graph.json","events_json":"https://pith.science/api/pith-number/J5S7YP7XGSAVB34LV4B3JFMZU3/events.json","paper":"https://pith.science/paper/J5S7YP7X"},"agent_actions":{"view_html":"https://pith.science/pith/J5S7YP7XGSAVB34LV4B3JFMZU3","download_json":"https://pith.science/pith/J5S7YP7XGSAVB34LV4B3JFMZU3.json","view_paper":"https://pith.science/paper/J5S7YP7X","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.06913&json=true","fetch_graph":"https://pith.science/api/pith-number/J5S7YP7XGSAVB34LV4B3JFMZU3/graph.json","fetch_events":"https://pith.science/api/pith-number/J5S7YP7XGSAVB34LV4B3JFMZU3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/J5S7YP7XGSAVB34LV4B3JFMZU3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/J5S7YP7XGSAVB34LV4B3JFMZU3/action/storage_attestation","attest_author":"https://pith.science/pith/J5S7YP7XGSAVB34LV4B3JFMZU3/action/author_attestation","sign_citation":"https://pith.science/pith/J5S7YP7XGSAVB34LV4B3JFMZU3/action/citation_signature","submit_replication":"https://pith.science/pith/J5S7YP7XGSAVB34LV4B3JFMZU3/action/replication_record"}},"created_at":"2026-05-18T01:04:04.827156+00:00","updated_at":"2026-05-18T01:04:04.827156+00:00"}